Subexponential parameterized algorithms and kernelization on almost chordal graphs

We study the algorithmic properties of the graph class Chordal-ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. We discover that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from Chordal-ke. We identify a large class of optimization problems on Chordal-ke that admit algorithms with the typical running time 2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on Chordal-ke when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on Chordal-ke graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on Chordal-ke graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of Chordal-ke, namely, Interval-ke and Split-ke graphs.


Introduction
Many NP-hard graph optimization problems are solvable in polynomial or even linear time when the input of the problem is restricted to a special graph class.For example, the chromatic number of a perfect graph can be computed in polynomial time [35], the Feedback Vertex Set problem is solvable in polynomial time on chordal graphs [31], and Hamiltonicity on interval graphs [45].From the perspective of parameterized complexity, the natural question here is how stable are these nice algorithmic properties of graph classes subject to some perturbations.For example, if an input n-vertex graph G is not chordal, but can be turned into a chordal graph by adding at most k edges, how fast can we solve Feedback Vertex Set on G? Can we solve the problem in polynomial time for constant k? Or maybe for k = log n or even for k = poly(log n)?
The problem of determining whether a graph G belongs to Chordal − ke, that is checking whether G can be turned into a chordal graph by adding at most k edges, is known in the literature as the Minimum Fill-in problem.The name fill-in is due to the fundamental problem arising in sparse matrix computations which was studied intensively in the past [55,57].The survey of Heggernes [37] gives an overview of techniques and applications of minimum and minimal triangulations.
Minimum Fill-in (under the name Chordal Graph Completion) was one of the 12 open problems presented at the end of the first edition of Garey and Johnson's book [30] and it was proved to be NP-complete by Yannakakis [63].Kaplan et al. proved that Minimum Fill-in is fixed parameter tractable by giving an algorithm of running time 16 k • n O (1) in [43].There was a chain of algorithmic improvements resulting in decreasing the constant in the base of the exponents [44,11,7] resulting with a subexponential algorithm of running time 2 O( √ k log k) • n O (1)  [28].A significant amount of work in parameterized algorithms is devoted to recognition problems of classes F − ke, F + ke, F − kv, and F + kv for chordal graphs and various subclasses of chordal graphs [1,2,4,6,15,13,14,26,42,51,62].
Parameterized algorithms, mostly for graph coloring problems, were studied on perturbed chordal graphs and subclasses of this graph class [12,59].Among other results, Cai [12] proved that Coloring (the problem of computing the chromatic number of a graph) is FPT (parameterized by k) on Split − ke graphs.Marx proved that Coloring is FPT on Chordal + ke and Interval + ke graphs but is W [1]-hard on Chordal + kv and Interval + kv graphs.Jansen and Kratsch [40] proved that for every fixed integer d, the problems d-Coloring and d-List Coloring admit polynomial kernels on the parameterized graph classes Split + kv, Cochordal + kv, and Cograph + kv.
Liedloff, Montealegre, and Todinca [49] gave a general theorem establishing fixed-parameter tractability for a large class of optimization problems.Let C poly be a class of graphs having at most poly(n) minimal separators.(Since every chordal graph has O(n) minimal separators, the class of chordal graphs is a subclass of C poly .)Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula, G be a graph, and t ≥ 0 be an integer.Liedloff, Montealegre, and Todinca proved that on graph class C poly + kv, the following generic problem Max |X| subject to There is a set F ⊆ V (G) such that X ⊆ F ; The treewidth of G[F ] is at most t; is solvable in time O(n O(t) • f (t, ϕ, k)), and thus is fixed-parameter tractable parameterized by k.The problem generalizes many classical algorithmic problems like Maximum Independent Set, Longest Induced Path, Maximum Induced Forest, and different packing problems, see [27].
Since the class C poly + kv contains Chordal − ke, the work of Liedloff et al. [49] yields that all these problems are fixed-parameter tractable on Chordal − ke graphs parameterized by k + t + |ϕ|.However, the theorem of Liedloff et al. cannot be used to derive our results.First, this theorem provides FPT algorithm only for problems of finding an induced subgraph of constant treewidth, which is not the case in our situation.Second, even for graphs of treewidth 0, their technique does not derive parameterized algorithms with subexponential running times.
Organization of the paper.The remaining part of the paper is organized as follows.In Section 2, we introduce notation and provide some useful auxiliary results.In Section 3, we discuss subexponential algorithms on Chordal − ke.Section 4 contains the lower bounds for Coloring and Clique on Chordal − ke.Sections 5-7 are devoted to kernelization.In Section 5, we give lower bounds and construct a polynomial Turing kernel for Weighted Clique on Chordal − ke.In Sections 6 and 7, we construct polynomial kernels for Independent Set on Interval − ke and Split − ke respectively.We conclude in Section 8 by some open problems.

Preliminaries
Graphs.All graphs considered in this paper are assumed to be simple, that is, finite undirected graphs without loops or multiple edges.For each of the graph problems considered in this paper, we let n = |V (G)| and m = |E(G)| denote the number of vertices and edges, respectively, of the input graph G if it does not create confusion.For a set X ⊆ V (G), X  2 denotes the set of pairs of distinct vertices of X.For a graph G and a subset X ⊆ V (G) of vertices, we write G[X] to denote the subgraph of G induced by X.We write G − X to denote the subgraph of G induced by V (G) \ X, and we write G − u instead of G − {u} for a single element set.Similarly, for an edge set A, G − A denotes the graph G = (V (G), E(G) \ A), and for a set of pairs of vertices , G + A is the graph G = (V (G), E(G) ∪ A).For a single-element set {e}, we use G+e for G+{e}.For A ⊆ V (G) 2 , G A denotes the graph G = (V (G), E(G) A).For a vertex v, we denote by N G (v) the (open) neighborhood of v, i.e., the set of vertices that are adjacent to v in G.
The complement of a graph G is the graph G with V (G) = V (G) such that two distinct vertices are adjacent in G if and only if they are not adjacent in G.A (proper) -coloring of a graph G is an assignment c : V (G) → {1, . . ., } of colors 1, . . ., to the vertices of G in such a way that adjacent vertices get distinct colors; a graph G is -colorable if it has an -coloring.
For a graph class C and a nonengative integer k, C − ke denotes the class of all graphs G such that there is a set A ⊆ V (G) 2 \ A of size at most k such that G + A ∈ C. In words, this means that C − ke contains graphs that can be turned to be graphs of C by at most k edge additions.

Graph classes.
A graph G is chordal (or triangulated ) if it does not contain an induced cycle of length at least four.In other words, every cycle of length at least four has a chord, i.e., an edge whose end-vertices are nonconsecutive vertices of the cycle.The intersection graph of a family of intervals of the real line is called an interval graph; it is also said that G is an interval graph if there is a family of intervals (called interval model or representation) such that G is isomorphic to the intersection graph of this family.A graph G is said to be split if its vertex set could be partition into independent set and a clique.We refer to [9,34] for detailed introduction to these graph classes.Notice that interval and split graphs are chordal.
A triangulation (or a chordal complementation) of a graph G is its chordal supergraph or, in other words, fill-in is the minimum number of edges whose addition makes the graph chordal.
and it is well-known that it is NP-hard to compute fill-in(G) and int-comp(G) [30] and the same holds for split-comp(G) [53].
We will make use of the following observation.
In particular, this observation implies that complexity lower bounds obtained for graph problems parameterized by the clique completion size hold for the same problems when they are parmeterized by the interval or split completion or by the fill-in, and the hardness for the interval or split completion parameterization implies the hardness for the fill-in parameterization.

Proposition 1 ([52]
).There is a polynomial algorithm that, given a graph G and a nonnegtive integer k, either correctly reports that fill-in(G) > k or returns a triangulation of G of size at most 8k 2 .
Tree decompositions.A tree decomposition of a graph G is a pair T = (T, {X t } t∈V (T ) ), where T is a tree whose every node t is assigned a vertex subset X t ⊆ V (G), called a bag, such that the following three conditions hold: In other words, every vertex of G is in at least one bag.
(T2) For every uv ∈ E(G), there exists a node t of T such that bag X t contains both u and v.
(T3) For every u ∈ V (G), the set T u = {t ∈ V (T )|u ∈ X t }, i.e., the set of nodes whose corresponding bags contain u, induces a connected subtree of T .
To distinguish between the vertices of the decomposition tree T and the vertices of the graph G, we will refer to the vertices of T as nodes.By the classical result due to Buneman and Gavril [10,32], every chordal graph G has a tree decomposition such that each bag of the decomposition is a maximal clique of G.Such a tree decomposition is referred as a clique tree of the chordal graph G.
It is more convenient to describe dynamic programming algorithms on tree decompositions of special nice form.We think of nice tree decompositions as rooted trees.That is, for a tree decomposition (T, {X t } t∈V (T ) ) we distinguish one vertex r of T which will be the root of T .We say that such a rooted tree decomposition (T, {X t } t∈V (T ) ) is nice if the following conditions are satisfied: • X r = ∅ and X = ∅ for every leaf of T .In other words, all the leaves as well as the root contain empty bags.
• Every non-leaf node of T is of one of the following three types: -Introduce node: a node t with exactly one child t such that X t = X t ∪ {v} for some vertex v / ∈ X t ; we say that v is introduced at t.
-Forget node: a node t with exactly one child t such that X t = X t \ {w} for some vertex w ∈ X t ; we say that w is forgotten at t.
-Join node: a node t with two children Throughout the paper, given a nice tree decomposition (T, {X t } t∈V (T ) ) of a graph G, we denote by V t the union of the bags in the subtree of T rooted in a node t ∈ V (T ).We will be using the following proposition, see e.g.[50].
Proposition 2. Every n-vertex chordal graph G admits a nice tree decomposition T = (T, {X t } t∈V (T ) ) such that every bag X t of G is a clique.This decomposition has at most O(n 2 ) nodes and can be constructed in polynomial time.
Parameterized Complexity and Kernelization.We refer to the books [17,21,25] for the detailed introduction to the field.Here we only briefly review the basic notions.Parameterized Complexity is a two dimensional framework for studying the computational complexity of a problem.One dimension is the input size n and the other is a parameter k associated with the input.
A parameterized problem is said to be fixed parameter tractable (or FPT) if it can be solved in time f (k) • n O (1) for some function f .Parameterized complexity theory also provides tools to disprove the existence of FPT algorithms under plausible complexity-theoretic assumptions.Here we use complexity lower bounds based on Exponential Time Hypothesis (ETH) formulated by Impagliazzo, Paturi and Zane [38,39].For an integer k ≥ 3, let q k be the infimum of the real numbers c such that the k-Satisfiability problem can be solved in time O(2 cn ), where n is the number of variables.Exponential Time Hypothesis states that δ 3 > 3.In particular, ETH implies that k-Satisfiability cannot be solved in time 2 o(n) n O (1) .
A compression of a parameterized problem Π 1 into a (non-parameterized) problem Π 2 is a polynomial algorithm that maps each instance (I, k) of Π 1 with the input I and the parameter k to an instance I of Π 2 such that (i) (I, k) is a yes-instance of Π 1 if and only if I is a yes-instance of Π 2 , and The output I is also called a compression.The function f is said to be the size of the compression.A compression is polynomial if f is polynomial.A kernelization algorithm for a parameterized problem Π is a polynomial algorithm that maps each instance (I, k) of Π to an instance (I , k ) of Π such that (i) (I, k) is a yes-instance of Π if and only if (I , k ) is a yes-instance of Π, and Respectively, (I , k ) is a kernel and f is its size.A kernel is polynomial if f is polynomial.
While it can be shown that every decidable parameterized problem is FPT if and only if it admits a kernel, it is unlikely that every problem in FPT has a polynomial kernel (see, e.g., [25] for the details).Still, even if a paramterized problem admits no polynomial kernel up to some complexity conjectures, sometimes we can reduce it to solving of a polynomial number of instances of the same problem such that the size of each instance is bounded by a polynomial of the parameter.
Let Π be a parameterized problem and let f : Z + → Z + be a computable function.A Turing kernelization or Turing kernel for Π of size f is an algorithm that decides whether an instance (I, k) of Π is a yes-instance in time polynomial in |I| + k, when given access to an oracle that decides whether (I , k ) is a yes-instance of Π in a single step if It is typical to describe a compression or kernelization algorithm as a series of reduction rules.A reduction rule is a polynomial algorithm that takes as an input an instance of a problem and output another, usually reduced, instance.A reduction rule is safe if the input instance is a yes-instance if and only if the output instance is a yes-instance.
In our paper we consider kernelization/compression for weighted problems.Because we are using weights, we have to compress their values as well.For this, we follow the approach proposed by Etscheid et al. [23] that is based on the algorithm for compressing numbers given by Frank and Tardos in [29].We state the result of Frank and Tardos in the form given in [23].

Proposition 3 ([29]
).There is an algorithm that, given a vector w ∈ Q h and an integer N , in polynomial time finds a vector w

Subexponential algorithms for induced d-colorable subgraphs
To construct subexponential algorithms on Chordal − ke, we consider tree decompositions such that each bag is "almost" a clique.Definition 1.Let k be a nonnegative integer.A tree decomposition T = (T, {X t } t∈V (T ) ) of a graph G is k-almost chordal if for every t ∈ V (T ), c-comp(G[X t ]) ≤ k, that is, every bag can be converted to a clique by adding at most k edges.
Note that every chordal graph has 0-almost chordal tree decomposition.Given a k-almost chordal tree decomposition, we are able to construct dynamic programming algorithms that are subexponential in k for various problem.The crucial property of the graphs in Chordal − ke is that we are able to construct a nice k-almost chordal tree decompositions for them in subexponential in k time by making use of the following result of Fomin and Villanger [28].
Proposition 4 ( [28]).Deciding whether graph G is in Chordal − ke can be done in time is of size at most k.Using Proposition 2, we construct in polynomial time a nice tree decomposition T = (T, {X t } t∈V (T ) ) of H with at most n 2 bags such that every bag X t is a clique of H. Clearly, T is a nice tree decomposition of G and G The crux of our subexponential algorithms is in the following combinatorial lemma.Lemma 2. Let d ≥ 1 be an integer.Let G be an n-vertex graph and let F be a set of induced d-colorable subgraphs of G. Let U ⊆ V (G) be a set of vertices of G such that c-comp(G[U ]) ≤ k, that is, U can be made a clique by adding at most k edges.Then

and
• the size of the projection of F on U , that is, the size of the family d) .Moreover, there is an algorithm that in time outputs a family of sets S ⊇ S, each set from S has at most 3d+ Proof.We partition U into sets X and Y as follows.Let X be the vertices of U that have at least one non-neighbor in U .In other words, for every v ∈ X there is u ∈ U that is not adjacent to v. Two observations about set X will be useful.First, because U , and hence X, can be turned into a clique by adding at most k edges, we have that |X| ≤ 2k.Second, the remaining vertices of U , namely, Y = U \ X, form a clique.For every set S ∈ S, we define S X = X ∩ S and can be turned into complete graph by adding at most k edges.These two conditions are used to estimate x.Let us remind that Turán graph is the complete d-partite graph on x vertices whose partition sets differ in size by at most 1.According to Turán's theorem, see e.g.[20], Turán graph has the maximum possible number of edges among all d-colorable graphs.The number of edges in Turán's graph is at most 1 2 x 2 d−1 d .Thus, Therefore, We obtain that which proves the first claim of the lemma.
To prove the second claim, we construct S as the family of all possible sets S of the form S X ∪ S Y , where S X ⊆ X is of size at most d+ and S Y ⊆ Y is of size at most d.Then every S ∈ S is also in S .To enumerate all sets from S , we brute-force through all subsets S X of X of size at most d+ ) subsets of X (if k ≥ 1) and it takes time k O(d+ √ dk) .Then in time n O(d) we brute-force through all subsets S Y of Y of size at most d.For each union S X ∪ S Y , we check in time 2 O(d+ √ kd) whether G[S X ∪ S Y ] is d-colorable using one of the exact algorithms [5,47] for this problem, and then either put it into S or discard.Thus the size of S is 2 O((d+ √ dk) log k) • n O(d) and the total time required to enumerate the sets of S is Let G be a graph and let F be an induced d-colorable subgraph of G. Informally, Lemma 2 says that for a given a k-almost chordal tree decomposition, every bag of this tree decomposition contains roughly O(d+ √ dk) vertices of F .This statement combined with dynamic programming over the tree decomposition could easily bring us to the algorithm computing a maximum dcolored subgraph of G in time n O(d+ √ dk) .However, this is not what we are shooting for; such an algorithm is not fixed-parameter tractable with parameter k.This is where the second part of the lemma becomes extremely helpful.Let us look at the family of all d-colorable induced subgraphs of G. Then the number of different intersections of the graphs from this family with a single bag of the tree decomposition is bounded by 2 O(d+ √ dk) n O(d) .This allows us to bound the number of "partial solutions" in the dynamic programming, which in turn brings us to a parameterized subexponential algorithm.As an example of the applicability of Lemma 2, we give an algorithm for the following generic problem.

Weighted d-colorable Subgraph
For d = 1, this is the problem of finding an independent set of maximum weight, the Weighted Independent Set problem.For d = 2, this is the problem of finding an induced bipartite subgraph of maximum weight, Weighted Bipartite Subgraph.
More precisely, we prove the following theorem.
Theorem 1.Let d ≥ 1 be an integer.For a given n-vertex graph G with a nice k-almost chordal tree decomposition with n O(1) bags, the 1) .We perform dynamic programming over T .Let us note that the width of the decomposition can be of order of n.The proof of the correctness for this dynamic programming is very similar to the one provided normally for graphs of bounded treewidth.However the running time analysis is based on Lemma 2.
Recall that then T is rooted at some node r, and for a node t of T , V t denotes the union of all the bags present in the subtree of T rooted at t, including For every node t, every S ⊆ X t such that G[S] is d-colorable, every mapping c : S → {1, . . ., d} of G[S], we define the following value: cost[t, S, c] = maximum possible weight of a set S such that (1) If c is not a proper coloring of G[S] or if no such set S exists, then we put cost[t, S, c] = −∞.We also put cost[t, ∅, c] be the maximum possible weight of a set S such that S ⊆ V t , S ∩ X t = ∅, and G[ S] is d-colorable.Then cost[r, ∅, c] is exactly the maximum weight of a d-colorable induced subgraph in G; this is due to the fact that V r = V (G) and X r = ∅.
Leaf node.If t is a leaf node, then we have cost[t, ∅, c] = 0.In this case, because the leaf node is an empty bag, the correctness of the formula is trivial.
Otherwise, we claim that the following formula holds: Here c is the coloring of S \ {v} extendible to c.
Proof.When v ∈ S, the formula trivially holds.Suppose that v ∈ S. Let S be a set maximizing cost[t, S, c].Because coloring c of S \ {v} is extendible to coloring c of S, which in turn is extendible to a d-coloring of S, we have that set S \ {v} is one of the sets considered in the On the other hand, let S be a set for which the maximum is attained in the definition of cost[t , S \{v}, c ].By the property of tree decompositions, all neighbors of v in V t , are contained in the bag X t .This yields that set S ∪ {v} induces a graph whose d-coloring can be obtained by extending coloring c of S, therefore, cost[t, S, c] ≥ w( For the running time required to compute (2).For fixed S and c, we have to verify whether c is a proper coloring of S, which can be done in time O(|S| 2 ) by going through all adjacencies of G[S].
Forget node.Let t be a forget node with child t such that X t = X t \ {w} for some w ∈ X t .Let S be any subset of X t ; again we assume that c is a proper d-coloring of G[S], since otherwise we put cost[t, S, c] = −∞.We claim that the following formula holds: On the other hand, let S be a set for which the maximum is attained in the definition of cost For running time, to compute (3), one has to go through all colorings c of S ∪ {w} which are extensions of coloring c of S. Thus for every possible color i of w, we have to check if this color is compatible with the coloring by c the neighbors of w in S.This can be done in time O(|S|).
Join node.Finally, suppose that t is a join node with children Let S be any subset of X t ; as before, we can assume that c is a proper d-coloring of G[S].The claimed recursive formula is as follows: Claim 3.3.Formula (4) is correct.
Proof.Let S be a set for which the maximum is attained in the definition of cost[t, S, c] and On the other hand, the union of sets maximizing the costs of cost[t 1 , S, c] and cost[t 2 , S, c], is a set whose d-coloring can be obtained from extending coloring c of S. Thus The running time to compute (4) for fixed set S and coloring c, is proportional to the time required to checked whether c is a proper coloring.Again, this can be done in time O(|S| 2 ).This concludes the description and the proof of correctness of the recursive formulas for computing the values of cost[•, •, •].The optimal subgraph of the maximum weight can be found by the standard backtracking arguments.Let us now estimate the total running time.The running time of our dynamic programming algorithm, up to a multiplicative factor O(|S| 2 ), is dominated by the number of triples [t, S, c].The number t is in n O (1) .Every set S should induce a d-colorable subgraph, so we can restrict our attention only to sets of the form X t ∩ V (F ) for some d-colorable graph F .By Lemma 2, each of these sets is of size at most d + d+ and the total number of such sets S for each bag . Hence the total running time of the dynamic programming is Combining Lemma 1 and Theorem 1, we immediately obtain the following corollary.
By Corollary 1, we immediately derive the following.In the Weighted Vertex Cover, we are given a weighted graph G and the task is to find a vertex cover of minimum weight, that is, a set of vertices X such that every edge of G has at least one endpoint in G. Similarly, in the Weighted Odd Cycle Transversal, we are asked to find a set of vertices of minimum weight such that every cycle of odd length contains at least one vertex from the set.Since the complement of every independent set is a vertex cover, and the complement of every induced bipartite subgraph is an odd cycle transversal, we have the following corollary.

Various extentions
The technique developed to prove Theorem 1 could be used to obtain subexponential algorithms for other problems beyond Weighted d-colorable Subgraph.These algorithms are very similar to the one from Theorem 1 and we sketch here only few problems and the adjustments required to make the dynamic programming from Theorem 1 to be applicable to these problems.
A homomorphism G → H from a graph G to a graph H is a mapping from the vertex set of G to that of H such that the image of every edge of G is an edge of H.In other words, a homomorphism G → H exists if and only if there is a mapping g : V (G) → V (H), such that for every edge uv ∈ E(G), we have g(u)g(v) ∈ E(H).Since there is a homomorphism from G to a complete graph K d on d vertices if and only if G is d-colorable, the deciding whether there is a homomorphism from G to H is often referred as the H-coloring.Note that if G admits an The main difference between solving Weighted H-colorable Subgraph, the problem of finding the maximum weight induced subgraph admiting a homomorphism to H, with Theorem 1 is that the value cost[t, S, c] in (1) should be redefined.In order to find a maximum weight Hcolorable induced subgraph of a graph G, we need to compute cost[t, S, g] = maximum possible weight of a set S such that . Similar running times could be derived for the variants of Weighted d-colorable Subgraph where some additional constrains on the properties of the d-colorable induced subgraph of minimum weight are imposed by some property C. For example, property C could be that the required subgraph is connected, acyclic, regular, degenerate, etc.As far as the information of the partial solution required for property C is characterized by set S ⊆ V t and all possible subsets of S or all permutations of S, we can solve the corresponding problem in time As a concrete example, consider Weighted d-degenerate Subgraph, whose task is to find a maximum weight d-degenerate induced subgraph H of the input graphs G. Recall that a graph is d-degenerate for a nonnegative integer d, if every induced subgraph of G has a vertex of degree at most d.In particular, every forest is 1-degenerate.An equivalent way of defining d-degenerate graph is in terms of coloring orderings [22].A vertex ordering of a graph is a dcoloring ordering, if each vertex has at most d neighbors that are after it in the ordering.Then a graph G is d-degenerate if and only if it admits an d-coloring ordering.In particular, this immediately implies that every d-degenerate graph is (d + 1)-colorable, that is, the chromatic number bound is automatically given by the d-degeneracy property.
Let G be a graph and let π be an ordering of its vertices.Let X ⊆ V (G).We say that π as an extension of an ordering π of X if the vertices of X occur in π in the same order as in π .Suppose that X = {x 1 , . . ., x r } and the vertices of X are indexed with respect to their order in π.Denote by δ π (G, X) = (δ 1 , . . ., δ r ) the sequence of nonnegative integers such that for each i ∈ {1, . . ., k}, δ i is the degree of x i in the graph obtained from G by the deletion of the vertices that occur before x i in π.Notice that if π is a d-coloring ordering of G, then δ i ≤ d for i ∈ {1, . . ., r}.We use the following lemma to construct our algorithm for Weighted d-degenerate Subgraph.
Then G has a d-coloring ordering if and only if there is an ordering π of X and orderings π 1 and π 2 of V 1 and V 2 respectively such that (i) π i is an extension of π for i = 1, 2, i − δ i ≤ d for every i ∈ {1, . . ., r}.Proof.Suppose that π is a d-coloring ordering.Then we define π, π 1 and π 2 as the orderings of X, V 1 and V 2 respectively as the ordering inherited from π, that is, every two vertices of the corresponding sets occur exactly in the same order as in π.It is straightforward to verify that (i) and (ii) are fulfilled.For the opposite direction, assume that there are π, π 1 and π 2 satisfying (i) and (ii).Let π = (x 1 , . . ., x r ).We construct the ordering π of G as follows.First, we concatenate the suborderings of π 1 and π 2 containing the vertices that occur before x 1 in these orderings and add x 1 in the end.Then for i = 2, . . ., r, we consecutively add suborderings of π 1 and π 2 composed by the vertices that occur between x i−1 and x i and add x i in the end.Finally, we add suborderings of π 1 and π 2 with the vertices that are after x r .It is easy to see that π is a d-coloring ordering of G because of (i) and (ii).
Using Lemma 3, we can define the values computed by our dynamic programming algorithm for Weighted d-degenerate Subgraph.For a nonnegative integer r, we define is the set of all sequences of length r of nonnegative integers at most d.Similarly to (1), for every node t, every S ⊆ X t such that G[S] is (d + 1)-colorable, every ordering π of S and every δ ∈ ∆ |S| , we define cost[t, S, π, δ] = maximum possible weight of a set S such that Many natural problem can be described by combining connectivity, degeneracy or degree constraints.Examples of such problems are various maximization problems like Weighted Induced Tree, Weighted Induced Path, Weighted Induced Cycle, as well as various packing variants of these problems like Weighted Induced Cycle Packing, the problem of finding a maximum induced subgraph whose each connected component is a cycle.
We summarize these observations with the following theorem.
Theorem 2. Let d ≥ 1 be an integer and G be an n-vertex graph from Chordal − ke.Then • Weighted Induced Tree, • Weighted Induced Path (Cycle), • Weighted Induced Cycle Packing, Another extension of Theorem 1 can be derived from the very recent results of Baste, Sau and Thilikos [3] about the F-Minor Deletion problem on graphs of bounded treewidth.Recall that a graph F is a minor of G if the graph isomorphic to F can be obtained from G by vertex and edge deletions and edge contractions.Respectively, G is said to be F -minor free if G does not contain F as a minor.For a family of graphs F, G is F-minor free if G is F -minor free for every F ∈ F. For a family F, the task of F-Minor Deletion is, given a graph G, to find a minimum set of vertices X such that G − X is F-free.Clearly, F-Minor Deletion is equivalent to F-Minor Free Induced Subgraph, whose task is to find a maximum F-minor free induced subgraph of G.A family of graphs F is connected if every F ∈ F is a connected graph.Baste et al. [3] obtained, in particular, the following result.

Proposition 5 ([3]
).Let F be a finite connected family of graphs.Then F-Minor Deletion can be solved in time 2 O(w log w) • n O (1) on graphs of treewidth at most w. 1t is well-known (see, e.g., the book [54] for the inclusion relations between the classes of sparse graphs) that if F is a finite family, then there is a positive integer d such that every F-minor free graph is d-degenerate.This means that for a finite family F, F-minor free graphs are d-colorable for some constant d that depends on F only.This allows us to use Lemma 2 and then combine our approach from Theorem 1 with the techniques of Baste et al. [3].Using Lemma 5, we obtain the following theorem.
Theorem 3. Let F be a finite connected family of graphs.Let also G be an n-vertex graph from Chordal − ke.Then F-Minor Free Induced Subgraph (or, equivalently, F-Minor For example, this framework encompasses such problems as Induced Planar Subgraph or Induced Outerplanar Subgraph whose task is to find a maximum size subgraph that is planar or outerplanar respectively. With a small adjustment the dynamic programming could be applied to the problems with specific requirements on the complement of the maximum induced d-colored subgraph.For example, consider the following problem.A set of vertices S ⊆ V (G) is a connected vertex cover if S is a vertex cover and G[S] is connected.Then in the Weighted Connected Vertex Cover problem, we are given a graph G with a weight function w : V (G) → Z + and the task is to find a connected vertex cover in G of minimum weight.Similarly, Weighted Connected Feedback Vertex Set is the problem of finding a connected feedback vertex set of minimum weight.
The complement of every vertex cover is an independent set, that is a 1-colorable subgraph, and the complement of every feedback vertex set is a forest, hence 2-colorable subgraph.While now the connectivity constraints are not on the maximum induced subgraph but on its complement our previous arguments can be adapted to handle these problems.Proof sketch.The proof is similar to the proof of Theorem 1, so we only briefly sketch the main idea.We also give the sketch for Weighted Connected Vertex Cover only, solution to Weighted Connected Feedback Vertex Set is basically the same.
Let (G, w, W ) be an instance of Weighted Connected Vertex Cover.
We use Lemma 1 to find a nice k-almost chordal tree decomposition T = (T, {X t } t∈V (T ) ) of G with O(n 2 ) bags.Let r be the root of T and V t be the union of the bags in the subtree of T rooted in t.
The following claim is crucial for our algorithm.To show the claim, it is sufficient to observe that any set of vertices constructed by picking an arbitrary vertex from each connected component of G[S] is independent.Hence, the number of connected components of G[S] is upper bounded by the maximum size of an independent set in X t .By Lemma 2, any independent subset of X t has size at most 3+ √ 1+8k 2 and the claim follows.
Let t ∈ T .For a set S ⊆ X t , denote by P t (S) the set of all partitions of S such that every two vertices x, y ∈ S that are in the same component are in the same set of each partition; we assume that P t (∅) = {∅}.
Notice that S ⊆ V (G) is a connected vertex cover if and only if U = V (G) \ S is an independent set and G[S] is a connected graph.
For every t ∈ V (T ), every independent set U ⊆ X t and every P ∈ P t (S) for S = X t \ U , our algorithm computes the value cost[t, U, P ] that is the minimum possible weight of a set S ⊆ V t such that • every two vertices x, y ∈ S are in the same set of P if and only if they belong to the same connected component of G[ S].
We compute cost[t, U, P ] bottom-up for t ∈ V (T ) starting from the leaves.The computations are trivial for leaf nodes and are performed in a standard way for introduce, forget and join nodes.In this section, we discussed optimization problems but, in many cases, similar dynamic programming can be applied for counting problems.For example, we can compute the number of (inclusion) maximal independent sets, maximal bipartite subgraphs, minimal (connected) feedback vertex sets, minimal connected vertex covers in time 2 O( √

Beyond induced d-colorable subgraphs
In Section 3, among other algorithms, we gave a subexponential (in k) algorithm on Chordal− ke graphs that finds a maximum d-colorable subgraph.In particular, this also implies that for every fixed d, deciding whether a graph from Chordal − ke is d-colorable, can be done in time subexponential in k.In this section we show that two fundamental problems, namely, Coloring and Clique, while still being FPT, but unlikely be solvable in subexponential parameterized time.

Coloring Chordal − ke graphs
First, we consider Coloring whose task is, given a graph G and a positive integer , decide whether the chromatic number of G is at most , that is, if G is -colorable.Note that here is not a fixed constant as in Section 3 and may be arbitrarily large.
Cai [12] proved that Coloring is FPT (parameterized by k) on Split − ke graphs.We generalize his result by showing that Coloring is FPT on Chordal − ke.Our approach is based on the dynamic programming which is similar to the one we used in Section 3.
We need the following property of graph colorings.Let G be a graph.It is well-known that an -coloring of a graph G can be seen as a partition X of V (G) into at most independent sets formed by the vertices of the same color that are called color classes.We call X an -coloring partition of G.We also say that a partition X of V (G) into independent sets is a coloring partition of G. Let X = {X 1 , . . ., x } be a coloring partition of G and let U ⊆ V (G).We say that the partition X | U of U formed by nonempty sets U ∩ X 1 , . . ., U ∩ X is a projection of X on U ; we assume that X ∅ contains the unique element ∅.Proof.Let S be the set of end-vertices of the elements of A in C. Then C = C \ S is a clique of G such that each vertex v of C is adjacent to every vertex of C distinct from v. Therefore, for every coloring partition X of G, the vertices of C form single-element sets of the projection of X on C. Therefore, only the vertices of S may be included in nontrivial sets of the partitions.Since |S| ≤ 2k, there are at most (2k) 2k partitions of S into independent sets.Hence, there are at most (2k) 2k partitions of C. To enumerate all the partitions, we do the brute force enumeration of the partitions of S and verify for each partition, whether this is a partition of S into independents sets.Since |C| ≤ n, the enumeration can be done in 2 O(k log k) • n time.Lemma 4 implies that there are at most (2k) 2k projections of the coloring partitions of G on C and these projections can be enumerated in 2 O(k log k) • n time.This allows to construct a dynamic programming algorithm for Coloring.Proof.Since the algorithm is constructed along the same lines as the algorithms from Section 3, we only briefly sketch the main idea.
Let (G, ) be an instance of Coloring.By Lemma 1, one can construct a nice k-almost chordal tree decomposition T = (T, {X t } t∈V (T ) ) of G with O(n 2 ) bags.Let r be the root of T and recall that V t denotes the union of the bags in the subtree of T rooted in t.For t ∈ V (T ), let Y t be the family of all partitions of X t into independent sets.We put Y For every X ∈ Y t , we define We compute the table of values of Col t (X ) bottom-up for t ∈ V (V ) starting from the leaves.Clearly, G is -colorable if and only if Col r (∅) = true.
Leaf node.Computing the tables for leaves is trivial as the bags are empty and we assume that the empty graph is -colorable.
From now on we assume that a node t ∈ V (T ) has children and the tables are already constructed fro them.
Introduce node.Let t be an introduce node of T .Denote by t its child and assume that X t = X t ∪ {v} for v / ∈ X t .For X ∈ Y t , we define X − v as the partition of X t obtained from X either by the deletion of {v} is {v} is an element of X or by the deletion of v from a nontrivial set of X containing v. For every X ∈ Y t , we set Forget node.Let t be a forget node of T .Denote by t its child and assume that X t = X t \{v} for v ∈ X t .For X ∈ Y t , we define X + v ⊆ Y t to be the sets of all partitions X of X t into independent sets such that X = X − v, that is, every X ∈ X + v is either obtained by adding the single-element set {v} or by including v into one of the independent set.For every X ∈ Y t , we set Col t (X ) := Join node.Let t be a join node of T with children t 1 and The correctness of the computation of the tables of values of Col t (X ) for t ∈ V (T ) is proved by standard arguments.By Lemma 4, we have that each table has at most (2k) 2k = 2 O(k log k) elements.The same lemma together with the description of the computing the tables for the leave and the introduce, forget and join nodes implies that the computation of the table for every t ∈ V (T ) can be done in time 2 O(k log k) n.Therefore, the total running time is 2 O(k log k) • n O (1) .Now we show that it is unlikely that Coloring can be solved in subexponential in k time.For this, we show the complexity lower bound based on ETH.For this, we use the result of Komusiewicz and Uhlmann [46] for the auxiliary Triangle Cover problem that, given a graph G with n = 3p vertices, asks whether V (G) can be covered by p disjoint triangles, that is, by p copies of K 3 that are subgraphs of G.

Proposition 6 ([46]
).Unless ETH is false, Triangle Cover cannot be solved in time 2 o(n+m) • (n + m) O (1) even when the input restricted to the graphs without induced K 4 .Now we rule out the existence of a subexponential algorithm for Coloring on Chordal−ke parameterized by k.In fact, we show a stronger claim.Theorem 6. Coloring cannot be solved in time 2 o(k) n O (1) on graphs in Complete−ke unless ETH fails.
Proof.A set of cliques {C 1 , . . ., C } of a graph G is a clique cover if the cliques are pairwise disjoint and V (G) = ∪ i=1 C i .It is well-known that G is -colorable if and only if its complement G has a clique cover of size .Let G be a graph with n = 3 vertices that has no induced K 4 .Observe that G has a clique cover of size if and only if V (G) can be covered by cliques if size 3, that is, by triangles.Consider a graph G with n = 3 vertices such that G has no induced K 4 .We obtain that (G, ) is a yes-instance of Coloring if and only if G is a yes-instance of Triangle Cover.Observe that G ∈ Complete − ke if and only if G has at most k edges.Then the existence of an algorithm for Coloring running in time 2 o(k) n O (1) on graphs in Complete − ke would imply that Triangle Cover can be solved in time 2 o(k) n O(1) on graphs without induced K 4 with at most k edges.This is impossible unless ETH is false by Proposition 6.

Clique in Chordal − ke graphs
Now we consider the Clique problem that ask, given a graph G and a positive integer , whether G has a clique of size at least .We show that Clique is FPT on Chordal − ke when parameterized by k even for the weighted variant of the problem in Section 5 by demonstrating that the problem admits a Turing kernel.Here, we prove the following lower bound.Theorem 7. Clique cannot be solved in time 2 o(k) n O (1) on graphs in Complete − ke unless ETH fails.
Proof.Observe that a graph G has a clique of size at least if and only if G has an independent set of size at least .Recall that G ∈ Complete − ke if and only if G has at most k edges.It was shown by Impagliazzo, Paturi and Zane [39] that Independent Set cannot be solved in time 2 o(n+m) n O (1) on graphs with n vertices and m edges unless ETH fails.These observations immediately imply the claim of the theorem.
We proved that Coloring and Clique do not admit subexponential algorithms on Complete− ke, when parameterized by k, unless ETH fails.Note that by Observation 1, this means that these problems cannot be solved by subexponential algorithms on Chordal − ke as well unless ETH is false.

Kernelization on Chordal-ke
In this section we discuss kernelization of the problems considered in the previous section.
Jansen and Bodlaender in [41] and Bodlaender, Jansen and Kratsch in [8] proved that Weighted Independent Set, Weighted Vertex Cover, Weighted Bipartite Subgraph, Weighted Odd Cycle Transversal, Weighted Feedback Vertex Set and Clique do not admit a polynomial kernel parameterized by the size of the minimum vertex cover of the graph unless NP coNP/ poly.
We use the following observation.
Observation 2. If G has a vertex cover of size at most k, then split-comp(G) ≤ k 2 .
Proof.Let G be a graph with a vertex cover X of size at most k.
Observation 2 together with the results of [8,41] give the following proposition.
Proposition 7. Weighted Independent Set, Weighted Vertex Cover, Weighted Bipartite Subgraph, Weighted Odd Cycle Transversal, Weighted Feedback Vertex Set and Clique do not admit a polynomial in k kernel on Split − ke graphs unless NP coNP/ poly.
By Observation 1, these problems parameterized by k have no polynomial kernel on Chordal− ke as well unless NP coNP/ poly.These results do not refute the existence of polynomial Turing kernels.We show that Weighted Clique has such a kernel.The input of Weighted Clique contains a graph G together with a weight function w : V (G) → Z + and a nonnegative integer W , and the task is to decide whether G has a clique C of weight at least W .
Our kernelization algorithm uses the following well-known property of chordal graphs.
Proposition 8 ( [33,60]).An n-vertex chordal graph has at most n inclusion-maximal cliques and they can be listed in linear time.
Theorem 8. Weighted Clique on Chordal − ke parameterized by k admits a Turing kernel with at most 16k 2 vertices with size O(k 8 ).
Proof.Let (G, w, W ) be an instance of Weighted Clique.
We apply Proposition 1 to approximate the fill-in of G.If the algorithm reports that fill-in(G) > k, we report that G / ∈ Chordal − ke and stop.Assume that this is not the case.Then the algorithm returns a set A ⊆ V (G) 2 of size at most 8k 2 such that G = G + A is a chordal graph.We define X to be the set of vertices that are the end-vertices of the edges of A. Note that |X| ≤ 16k 2 .Then we use Proposition 8 to list all the inclusion maximal cliques Observe that (G, w, W ) is a yes-instance of Weighted Clique if and only if (G i , w, W i ) is a yes-instance for at least one i ∈ {1, . . ., r}.Then Turing kernelization algorithm solves Weighted Clique by calling the oracle for each instance (G i , w, W i ) with the parameter k for i ∈ {1, . . ., r}. Since . ., r}, we solve instances with the input graphs of bounded size.To be able to call the oracle that solves the instances of bounded size, we have to compress the weights.For this, we apply Proposition 3.
Since sign(w • b) = sign( w • b) for all vectors b ∈ Z h with b 1 ≤ N − 1, the equality holds for every b whose elements are in {−1, 0, 1}.This implies that the weights w(v j ) are positive and Wi is nonnegative.Also we have that for every set of indices I ⊆ {1, . . ., p}, j∈I w(v j ) ≤ W i if and only if j∈I w(v j ) ≤ Wi .This proves that the instances (G i , w, W i ) and (G i , w, Wi ) are equivalent.
Since |V (G i )| ≤ 16k 2 , we obtain that the weights in the instance (G i , w, Wi ) can be encoded by stings of length O(k 6 ) for i ∈ {1, . . ., r}.Hence, the size of each instance is O(k 8 ).
Propositions 1, 8 and 3 immediately imply that the kernelization algorithm runs in polynomial time.

Independent Set on Interval − ke
In this section we show that Independent Set parameterized by the size of interval completion of the input graph admits a polynomial compression into the Weighted Independent Set problem.We state Weighted Independent Set as a decision problem, whose input contains a graph G with a weight function w : V (G) → Z + and a nonnegative integer W , and the task is to decide whether G has an independent set S with w(S) ≥ W .More formally, we show the following theorem.
Theorem 9. Independent Set on G ∈ Interval − ke admits a compression of size O(k 56 ) into Weighted Independent Set.
Since Weighted Independent Set is in NP and, consecutively, has a polynomial reduction to Independent Set that is NP-complete [30], the theorem immediately gives the following corollary.The remaining part of the section contains the proof of Theorem 9.In Subsection 6.1, we introduce additional notions and state some auxiliary results.Then, in Subsection 6.2, we give the compression itself.

Technical lemmas
An interval graph has been defined as an intersection graph of a family of intervals of the real line but for our compression algorithm we need the characterization of interval graphs in terms of forbidden induced subgraphs.
Three pairwise nonadjacent vertices of a graph form an asteroidal triple (AT) if there is a path between every two of them that avoids the closed neighborhood of the third.For an asteroidal triple T of a graph G, a T -AT-witness is an inclusion minimal induced subgraph F of G such that T is an asteroidal triple of F .The vertices of T are called terminals of F .Clearly, F is induced by the vertices of induced paths between every two vertices of T that avoid the closed neighborhood of the third.This implies that the existence of an asteroidal triple T can be checked in polynomial time and then the construction of a T -AT-witness can be done in polynomial time using the self-reducibility technique.An asteroidal witness is an inclusion minimal induced subgraph that contain an asteroidal triple and we call the verices of an asteroidal triple terminals of the witness (note that the choice of terminals may be not unique).
It is said that a graph G is AT-free if G has no asteroidal triple.We use the following classical result of Lekkerkerker and Boland [48].

Proposition 9 ([48]
).A graph G is an interval graph if and only if G is chordal and AT-free.
The main result of Lekkerkerker and Boland [48] is the characterization of interval graphs by forbidden induced subgraphs.We use this characterization in the following form tailored for our purposes.

Lemma 5 ([48]
).If a chordal graph G contains an asteroidal triple T , then it contains a minimal asteroidal witness F isomorphic to one of the graphs F 1 , F 2 , F 3 (r) for r ≥ 2 or F 4 (r) for r ≥ 1 that are shown in Figure 1 (a)-(d) and for every nonterminal vertex v of F , it holds that v ∈ V (G) \ T .Moreover, such a witness F can be found in polynomial time.
The last statement of the lemma means that the vertices of an asteroidal triple of G may be only terminal vertices of a minimal witness.
We say that vertex set ) such that G + A is chordal.Note that if G + A is chordal then the set of endvertices of the edges of A is a chordal-complementing set.Observe also that every superset of a chordal-complementing set is chordal-complementing, and for each We say that F is associated with T .
Our compression algorithm uses the properties of chordal-complementing sets and X-touching ATs with associated witnesses given in the following two lemmas.Lemma 6.Let X ⊆ V (G) be a chordal-complementing set for a graph G and let T be an X-touching AT triple of G. Then G has an X-touching AT T with an associated witness F isomorphic to one of the graphs F 1 , F 2 , F 3 (r) for r ≥ 2, F 4 (r) for r ≥ 1 or F 5 that are shown in Figure 1 (a)-(e).Moreover, T and an associated witness F can be constructed in polynomial time.
Proof.Suppose that T = {u 1 , u 2 , u 3 } is an X-touching AT.We find an associated witness F .Note that this can be done in polynomial time using the self-reducibility technique.
Assume that F is not a chordal graph.Then it contains an induced cycle C with at least four vertices.Notice that since X is a chordal-complementing set, C contains at least two nonadjacent vertices from T .We consider four cases depending on the length of C.
Let C be of length 4. By symmetry, we assume without loss of generality that Recall that F is induced by the vertices of induced paths between every two vertices of T that avoid the closed neighborhood of the third.The vertices v 1 and v 2 cannot belong to any induced (u 1 , u 3 ) or (u 2 , u 3 )-path that avoids witness but this contradicts the minimality of F .We conclude that C has length at least 5.
Suppose that C has length 5. Again, we can assume without loss of generality that ∈ A, then C is a cycle of G .This contradicts the condition that X is a chordal-complementing set.Hence, C has length at least 6.
Assume that C has length 6. Suppose that |V (C)∩T | = 2. Then we can assume that u 1 , u 2 ∈ V (C) and either In both cases, we obtain a contradiction with the condition that X is a chordal-complementing set in the same way as in the previous case, because for every A ⊆ witness as required by the lemma.
Finally, let C be of length at least 7. Then for two vertices of T , say, u 1 and u 2 , C contains an induced (u 1 , u 2 )-path P of length at least 3. Then for every A ⊆ X 2 \ E(G[X]), G = G + A contains a cycle of length at least four that contains P as a segment.This contradicts the condition that X is a chordal-complementing set.
Assume now that F is a chordal graph.Then the claim of the lemma is a direct corollary of Lemma 5.
Lemma 7. Let X ⊆ V (G) be a chordal-complementing set for a graph G. Suppose that G has an X-touching AT T with an associated witness F isomorphic to one of the graphs F 1 , F 2 , F 3 (r) for r ≥ 2, F 4 (r) for r ≥ 1 or F 5 that are shown in Figure 1 (a)-(e).Then for every interval complementation H of G, H has an edge uv / ∈ E(G) such that Proof.Suppose that T is an X-touching AT with an associated witness isomorphic to one of the graphs F 1 , F 2 , F 3 (r) for r ≥ 2, F 4 (r) for r ≥ 1 or F 5 .Let H be an interval complementation of If the vertices of V (F ) ∩ T are nonadjacent in H, then the existence of uv / ∈ E(G) satisfying (i) and (ii) follows from Proposition 9. Let these vertices be adjacent.Suppose that z 1 , z 2 of F (see Figure 1) are in T .Observe that F + z 1 z 2 contains an induced cycle of length at least four as it is shown in Figure 2 (a)-(e).Since H is chordal, we obtain that there is uv ∈ E(H) \ E(G) satisfying (i) and (ii).The case z 2 , z 3 ∈ T is symmetric.Assume that z 1 , z 3 ∈ T .By symmetry, it is sufficient to consider the cases F = F 2 , F = F 3 (r) and F = F 4 (r).Again, we observe that F + z 1 z 3 contains an induced cycle of length at least four as it is shown in Figure 2  Then F + {z 1 z 2 , z 2 z 3 , z 1 z 3 } coincides with F 4 (2).Since H has no induced subgraph isomorphic to F 4 (2), we have that there is uv ∈ E(H) \ E(G) satisfying (i) and (ii).
In our compression algorithm, we have to compute a maximum independent set for chordal graphs.It was already observed by Gavril [31] in 1972 that this in polynomial (linear) time on chordal graphs.

Compression
In this section, we give a compression of Independent Set on G ∈ Interval − ke parameterized by k.Let (G, ) be an instance of Independent Set and let a nonnegative integer k be the parameter.
First, we apply the algorithm of Natanzon, Shamir and Sharan [52] (see Proposition 1) to approximate the fill-in of G.If the algorithm reports that fill-in(G) > k, we immediately stop as, clearly, G / ∈ Interval − ke.Assume that this is not the case.Then the algorithm returns a set A ⊆ V (G) 2 of size at most 8k 2 such that G + A is a chordal graphs.We define X to be the set of vertices that are the end-vertices of the edges of A. Note that X is a chordal-complementing set.Now we proceed with enhancing X. Reduction Rule 6.4.If G has an X-touching AT T with an associated witness F isomorphic to one of the graphs F 1 , F 2 , F 3 (r) for r ≥ 2, F 4 (r) for r ≥ 1 or F 5 that are shown in Figure 1 (a)-(e), then set X := X ∪ V (F ).
We apply the rule exhaustively but at most k + 1 times, because in the same way as for Reduction Rule 6.2, we can apply Lemma 7. Hence, the next rule is safe.Reduction Rule 6.5.If Reduction Rule 6.4 have been applied k + 1 times, then report that G / ∈ Interval − ke and stop.
Assume that the algorithm did not stop.By Observation 3, each F , whose vertices have been added to X by Reduction Rule 6.4, has at most 2p + 4 = 32k 2 + 2k + 4 vertices.Then By applying Reduction Rule 6.4, we achieve an important property of G and X.Since the rule cannot be applied any more, by Lemma 6, the following holds.
Next, we simplify the instance (G, ) of Independent Set by the removal chordal components.It is straightforward to see that the following rule is safe.
Reduction Rule 6.6.If G has a connected component H that is a chordal graph, then compute the size α(H) of a maximum independent set of H and set G := G − V (H), X := X \ V (H) and = − α(H).If ≤ 0, then return a trivial yes-instance of Weighted Independent Set and stop.
The rule is applied exhaustively.We assume that the algorithm did not stop.For a set Y ⊆ X of size at most two, let C Y be the set of connected components of the graph G − (X ∪ N G (Y ) and define Notice that C is a set of connected induced subgraphs of G − X and distinct subgraphs in the set can have common or adjacent vertices.For each C ∈ C, let I(C) be a maximum independent set of C. We use the following crucial property of C.
Lemma 9.There is a set S ⊆ X and a family C * ⊆ C of pairwise disjoint graphs without adjacent (in G) vertices in distinct subgraphs such that is a maximum independent set of G.
Proof.Let I be a maximum independent set of G. Let S = I ∩ X and denote by C 1 , . . ., C s the connected components of G − (X ∪ N G (S)).Clearly, I \ S ⊆ V (C 1 ) ∪ . . .∪ V (C s ).Moreover, if I i is an arbitrary maximum independent set of C i for i ∈ {1, . . ., s}, then I = S ∪ I 1 ∪ . . .∪ I s is a maximum independent set of G.We claim that for each i ∈ {1, . . ., s}, there is a set Y ⊆ S of size at most two such that C i ∈ C Y .
To obtain a contradiction, assume that there is i ∈ {1, . . ., s} such that C i / ∈ C Y for any Y ⊆ S of size at most two.Then |S| ≥ 3 and there are distinct vertices u 1 , u 2 , u 3 ∈ S such that for every j ∈ {1, 2, 3}, there is v j ∈ N G (u j ) \ X such that (i) v j / ∈ N G (u h ) for h ∈ {1, 2, 3} \ {j} and (ii) v j is adjacent to some vertex w j of C i .Consider any two distinct vertices u j and u h for j, h ∈ {1, 2, 3}.Let t be the unique element of {1, 2, 3} \ {j, h}.Since C i is connected, there is a (w j , w h )-path P in C i .Notice that P avoids N G [u t ].Let P = u j v j P v h u h .We have that P is a (u j , u h )-path in G = G − E(G[X]) that avoids the neighborhood of u t .Since this holds for any choice of j and h, we obtain that T = {u 1 , u 2 , u 3 } ⊆ X is an asteroidal triple T in G with a T -AT-witness F such that V (F ) ∩ X = T but this contradicts Observation 4. This proves that for each i ∈ {1, . . ., s}, there is a set Y ⊆ S of size at most two such that C i ∈ C Y .
We obtain that C 1 , . . ., C s ∈ C are pairwise disjoint graphs without adjacent vertices in distinct subgraphs and We show that C has size that is bounded by a polynomial of the parameter.First, we show an auxiliary claim.Proof.To obtain a contradiction, assume that |C | > 3p, that is, there are at least 3p + 1 If y is adjacent to at least p+1 connected components of G , then N G (y)\X contains an independent set of size at least p + 1 and we would be able to apply Reduction Rule 6.3; a contradiction.Therefore, each y ∈ N G (x) \ X has neighbors in at most p connected components of G .Since C ≥ 3p + 1, we conclude that there are three distinct vertices y 1 , y 2 , y 3 ∈ N G (x) \ X such that there are three distinct connected components C 1 , C 2 , C 3 ∈ C with the property that for every i ∈ {1, 2, 3}, y i has a neighbor z i ∈ V (C i ) and y i has no neighbor in C j for j ∈ {1, 2, 3} \ {i}.Consider H = G[{x, y 1 , y 2 , y 3 , z 1 , z 2 , z 3 }].It is easy to see that z 1 , z 2 , z 3 is an asteroidal triple in this graph and H contains an induced subgraph isomorphic to F 1 , F 3 (2) or F 3 (3) (see Figure 1 (a) and (c)) depending on the adjacencies between y 1 , y 2 and y 3 .This means that we would be able to apply Reduction Rule 6.4 contradicting the assumption that the rule was applied exhaustively.This proves that |C | ≤ 3p.Let Y = ∅.Observe that each connected component of G contains a vertex of X, because G has no connected component that is a chordal graph as Reduction Rule 6.6 cannot be applied.Note that if a vertex x has neighbors in at least p + 1 connected components of G − X for p = 8k 2 + 7k + 2, then N G (x) \ X contains an independent set of size at least p + 1 and we would be able to apply Reduction Rule 6.3.Therefore, each vertex of X has neighbors in at most p components of G − X.Hence, |C ∅ | ≤ p|X| = O(k 5 ) by the definition of p and (5).
Suppose that Y = {x} for x ∈ X. Denote by C (1) the set of graphs in C Y that have vertices adjacent to N G (x) \ X and let C (2) be the set of graphs in C Y that have no vertex adjacent to N G (x) \ X.We have that C Y = C (1) ∪ C (2) .By exactly the same arguments as for Y = ∅, we obtain that |C (2) | ≤ p(|X| − 1).By Lemma 10, |C (1) | ≤ 3p.We obtain that all vectors b ∈ Z h with b 1 ≤ N − 1.We define w * (v i ) = w i for i ∈ {1, . . ., s} and consider the instance (G * , w * , W ) of Weighted Independent Set.This completes the construction of the compression.The properties of the obtained instance of Weighted Independent Set are summarized in the following lemma.
Lemma 13.The instance (G * , w, ) is a yes-instance of Weighted Independent Set if and only if (G * , w * , W ) is a yes-instance.The size of (G * , w * , W ) is O(k 56 ).
Proof.Notice that, in particular, the equality sign(w • b) = sign( w • b) holds for all vectors b ∈ Z h such that each element of b is −1, 0 or 1.This implies that the elements of w are positive and for every J ⊆ {1, . . ., s}, i∈J w(v i ) ≥ if and only if i∈J w i ≥ W . Clearly, for every set of vertices S ⊆ V (G * ), v∈S w(v) ≥ if and only if v∈S w * (v) ≥ W .This means that (G * , w, ) is a yes-instance of Weighted Independent Set if and only if (G * , w * , W ) is a yes-instance.
Since w ∞ ≤ 2 4h 3 N h(h+2) , we have that w(v) ≤ 2 4(s+1) 3 (s+2) (s+1)(s+3) for every v ∈ V (G * ) and the same upper bound holds for W .This implies that the weights of the Running time evaluation.Finally, we have to show that the compression algorithm is polynomial.The construction of the initial set X can be done in polynomial time by Proposition 1. Then we apply Reduction Rules 6.1-6.6.It is straightforward to see that Reduction Rule 6.1 can be applied in polynomial time as we are looking in it for an induced subgraph of bounded size.Reduction Rules 6.2 and 6.5 are trivial.Reduction Rules 6.3 and 6.6 are polynomial by Proposition 10.Reduction Rule 6.4 is polynomial by Lemma 6.Since Reduction Rules 6.1 and 6.4 are applied at most k + 1 times and Reduction Rules 6.3 and 6.6 are applied at most |V (G)| times, we conclude all the rules can be applied in polynomial time.In the next step, we construct the instance (G * , w, ) and the step is polynomial due to Proposition 10.Finally, we reduce the weight and this can be done in polynomial time by Proposition 3.This completes the proof of Theorem 9.

Independent Set on Split − ke
In this section, we show that Independent Set admits a polynomial kernel when parameterized by the split completion size.
It is known to be NP-hard to compute split-comp(G) [53] but, interestingly, if we allow not only edge additions but also edge deletions, then the problem becomes polynomial time solvable as it was proved by Hammer and Simeone in [36].

Proposition 11 ([36]
).There is an algorithm that, given a graph G, in polynomial time finds a set A ⊆ V (G) 2 of minimum size such that G A is a split graph.
Theorem 10.Independent Set on Split − ke admits a polynomial kernel with at most 2k 2 (k + 2) vertices when parameterized by k.
Proof.Let (G, ) be an instance of Independent Set and let a nonnegative integer k be the parameter.
We use Proposition 11 and find a set ∈ Split − ke and stop.Assume that is not the case.Let D = A ∩ E(G) and C = A \ E(G), that is, D is the set of deleted edges and C is the set of added edges.We find a partition of V (G ) into a clique K and an independent set I. Note that by the minimality of A, the edges of D have their end-vertices in I and the edges of C have their end-vertices in K. Let X be the set of end-vertices of C and set Y = K \ X.Note that |X| ≤ 2k and every vertex of X is adjacent to each vertex of Y .
We apply a series of reduction rules for (G, ) together with the sets I, X, Y, D.
Reduction Rule 7.1.If D = ∅, then for uv ∈ D do the following.
• If each of u and v is not adjacent to at least k + 2 vertices of Y in G, then report that G / ∈ Split − ke and stop.
• Otherwise, pick a vertex of {u, v} with the minimum number of nonneighbors in Y , say u, and We apply the rule exhaustively, until D becomes empty.
2 \ E(G) be a set of pairs of vertices of minimum size such that Ĝ = G + R is a split graph.Let (S, Z) be a partition of V ( Ĝ) into an independent set S and a clique Z.Notice that either u ∈ Z or v ∈ Z.By symmetry, assume without loss of generality that u ∈ Z. Observe that Y is a clique of G. Hence, |Y \ Z| ≤ 1.This implies that R contains at least k + 1 edges incident to u in Ĝ whose other end-vertices are in Y .Therefore, split-comp(G) = |R| ≥ k + 1. Claim 7.1 guarantees that if we stop by Reduction Rule 7.1, then G / ∈ Split − ke.Assume that the algorithm did not stop.Then we obtain that the constructed sets X, Y and I have the properties that are summarized in the following claim.
Claim 7.2.The sets X, Y, I form a partition of the vertices of G such that To see (i), it is sufficient to observe that only the edges of D had both their end-vertices in I in the initial I and we exclude at least one end-vertex of every edge of D from I by Reduction Rule 7.1.
The property (ii) is trivial as Y was a clique before we started to apply Reduction Rule 7.1 and we only delete vertices from Y by the rule.
For (iii), observe that if v is a vertex of the initial set X, then Y ⊆ N G (v) by the definition of X and Y .Then, if we add a vertex u ∈ I to Y by Reduction Rule 7.1, then we delete the vertices of Y \ N G (u) from Y .Note that these vertices are included in X and, since Y is a clique, they are adjacent to all remaining vertices of Y .Hence, Y ⊆ N G (v) for every v ∈ X.
Finally, to prove (v), observe that initially |X| ≤ 2|C| and, therefore, every independent set with its vertices in the initial set X has size at most 2|C|.By each application of Reduction Rule 7.1, we put a vertex u ∈ I in X and add a clique Y \ N G (u).This means that we can increase the maximum size of independent subset of X by at most two.The rule is applied at most |D| times.We conclude that the maximum size of independent subset of X is at most 2|C| + 2|D| = 2|A| ≤ 2k.Proof of Claim 7.3.Let G be the graph obtained from G by the application of Reduction Rules 7.2 and 7.3 and denote by I the set obtained from I. For the safeness proof, it is sufficient to show that if G has an independent set S with |S| ≥ , then G has an independent set of size at least .If S ∩ Y = ∅, S ⊆ V (G ), that is, S is an independent set of G .Suppose that there is v ∈ S ∩ Y .Since Y is a clique, v is the unique vertex of S in Y .Since Y ⊆ N G (u) for every u ∈ X, X ∩ S = ∅.This means that S \ {v} ⊆ I.
Suppose that there is u ∈ I such that v ∈ N G (u).Consider S = (S \ {v}) ∪ {u}.We obtain that S is an independent set, since S ⊆ I. To see that I constructed by Reduction Rule 7.3 is independent, it is sufficient to observe that we include in I a unique vertex of Y that is not adjacent to other vertices of I.
Observe that after applying Reduction Rules 7.2 and 7.3, we have that (X, I) is a partition of V (G), where I is an independent set.Reduction Rule 7.4.If there is a vertex u ∈ X such that |N G (u)| ≥ 2k, then set X := X \{u}.We apply the rule exhaustively.We obtain G with the property that |N G (u) ∩ I| ≤ 2k − 1 for every u ∈ X.
Finally, we exhaustively apply the following reduction rule.It is straightforward to see that the rule is safe.Now we can show that the obtained graph G has bounded size.This completes the construction of the kernel.The initial construction of X, Y , D and I is polynomial by Proposition 11.It is straightforward to see that Reduction Rules 7.1-7.5 can be applied in polynomial time.Hence, our kernelization algorithm is polynomial.

Conclusion
In this paper, we initiated the study of parameterized subexponential and kernelization algorithms on Chordal−ke graphs.The existence of such algorithms makes this graph class a very interesting object for studies.For other structural parameters, like treewidth or vertex cover, we have quite good understanding about the complexity of various optimization problems derived from general meta-theorems like Courcelle's or Pilipczuk's theorems [16,56] and advanced algorithmic techniques [19,18,24].We believe that further exploration of the complexity landscape of fill-in parameterization is an interesting research direction.If an optimization problem is NP-complete on chordal graphs, like Dominating Set, then on Chordal − ke this problem is in Para-NP.On the other hand, even if a problem is solvable in polynomial time on chordal graphs, in theory, there is nothing preventing it from being Para-NP on Chordal − ke.Is there a natural graph problem with this property?For many problems that are solvable in polynomial time on chordal graphs, we also established FPT algorithms on Chordal − ke class.This does not exclude a possibility that there are problems that are not FPT parameterized by k but solvable in polynomial time for every fixed k.We do not know any such problem (in other words, the problem in class XP) yet.It will be interesting to see, if there is any natural graph problem of such complexity.In addition, we proved that there are problems that are FPT on Chordal − ke when parameterized by k and which cannot be solved in subexponential time unless ETH fails.We believe it would be exciting to obtain a logical characterization of problems that can be solved in subexponential time on Chordal − ke when parameterized by k, similar to the classical Courcelle's theorem [16].Some concrete open problems.Observe that for our subexponential dynamic programming algorithms, we only need a k-almost chordal tree decomposition of the input graph, that is, a decomposition where each bag can be made a clique by adding at most k edges.(Recall Definition 1.) The maximum of numbers c-comp(G[X t ]) ≤ k can be significantly smaller than the minimum fill-in of a graph.For graphs in Chordal − ke, we can find fill-in in a subexponential in k time by the algorithm of Fomin and Villanger [28].However, we do not know if it is FPT in k to decide, whether a graph admits a k-almost chordal tree decomposition.And if yes, can it be done in subexponential time?
Finally, we proved that Independent Set on Interval − ke and Split − ke admit polynomial kernels when parameterized by k.We leave open the question whether or not this problem has a polynomial (Turing) kernel on Chordal − ke.
and we denote the minimum size of an interval (split, clique) complementation by int-comp(G) (split-comp(G), c-comp(G) respectively).Clearly, G has an interval (split, clique) complementation of with the completion size at most k if and only

√ d 2 +8dk 2 and
all subsets of Y of size at most d.Since |X| ≤ 2k, we go through at most (2k) O( d+ √ and c is a proper coloring of G[S] extendible to a proper d-coloring of G[ S].

Corollary 2 .
Weighted Independent Set and Weighted Bipartite Subgraph on G ∈ Chordal − ke are solvable in time 2 O( √ k log k) • n O(1) .

Corollary 3 .
Weighted Vertex Cover and Weighted Odd Cycle Transversal on Chordal − ke graphs are solvable in time 2 O( √ k log k) • n O(1) .
and g is a homomorphism from G[S] to H extendible to a homomorphism of G[ S] to H.The number of homomorphisms g from G[S] to H does not exceed |V (H)| |S| .The running time of operations join, introduce and forget operations in the dynamic programming can be bounded by a polynomial of the number of states.Hence we can solve the problem in time 2 O((|V (H)|+ and ordering π of S can be extended to a d-coloring ordering π of G[ S] with δ π (G[ S], S) = δ.For a given S ⊆ X t , there are |S|!different ordering of the vertices of S and |∆ |S| | = (d + 1) |S| .By Lemma 2, we have that |S| ≤ 3d+ √ d 2 +8dk 2 and, therefore, we consider 2 O((d+ √ dk) log(dk)) orderings π and 2 O((d+ √ dk) log d) sequences δ.Taking into account that by Lemma 2, we consider 2 O((d+ √ dk) log k) • n O(d) sets S, we obtain that the table of our dynamic programming algorithmstores 2 O((d+ √ dk) log k) •n O(d) values.Applying Lemma 3 and the standard dynamic programming arguments, it is easy to show that we can solve Weighted d-degenerate Subgraph using the information stored in the tables.

Theorem 4 .
Weighted Connected Vertex Cover and Weighted Connected Feedback Vertex Set are solvable in time 2 O( √ k log k) • n O(1) on Chordal − ke.

Claim 3 . 4 .
Let t be a node of T and let S ⊆ X t .Then G[S] has at most 3+ Finally, we compute cost[r, ∅, {∅}], which gives us the minimum weight of a connected vertex cover.By Lemma 2, X t has 2 O( √ k log k) n O(1) independent subsets.By Claim 3.4, we have that for every S ⊆ X t , G[S] has at most 3+ √ 1+8k 2 connected components.Therefore, |P t (S)| = 2 O( √ k log k) and the algorithm computes 2 O( √ k log k) •n O(1) values of cost[t, U, P ] for each t ∈ V (T ).Therefore, the running time of the algorithm is 2 O( √ k log k) • n O(1) .

Lemma 4 .
Let G ∈ Chordal − ke and let A ⊆ V (G) 2 such that |A| ≤ k and G = G + A is a chordal graph.Let also C be a clique of G .Then there are at most (2k) 2k partitions of C into independent sets and these partitions can be enumerated in 2 O(k log k) • n time.

Theorem 5 .
Coloring can be solved in time 2 O(k log k) • n O(1) on Chordal − ke.

Figure 2 :
Figure 2: Induced cycles in F shown by thick lines.
(f)-(h) and the claim follows.Let |V (F ) ∩ T | = 3, that is T = {z 1 , z 2 , z 3 }.If the vertices of V (F ) ∩ T are pairwise nonadjacent in H, then the existence of uv ∈ E(H) \ E(G) satisfying (i) and (ii) follows from Proposition 9.If H[T ] contains an edge, then we apply the same arguments as above for the cases F = F 2 , F = F 3 (r) and F = F 4 (r) and obtain that F + E(H[T ]) contains an induced cycle of length at least four.This implies that there is uv ∈ E(H) \ E(G) satisfying (i) and (ii).Let F = F 5 .If |E(H[T ])| = 1 or |E(H[T ])| = 2, we again have that F +E(H[T ]) contains an induced cycle of length at least four (see Figure 2 (e) and (i)) and the claim follows.Let |E(H[T ])| = 3.

Lemma 10 .
Let x ∈ Y ⊆ X, where |Y | ≤ 2, and let C ⊆ C Y be the set of graphs in C Y that have at least one neighbor in N G (x) \ X.Then |C | ≤ 3p for p = 8k 2 + 7k + 2.
and, therefore, |C Y | = O(k 5 ) by the definition of p and (5).
vertices and W can be encoded by a string of length O(k 42 ), because by Lemma 11, |V (G * )| = O(k 14 ).Because G * has O(k 14 ) vertices and O(k 28 ) edges, the size of (G * , w * , W ) is O(k 56 ).

Claim 7 . 1 .
If the algorithm stops while executing Reduction Rule 7.1, then split-comp(G) > k.Proof of Claim 7.1.The algorithm stops if there is uv These properties allow us to apply the next two rules.Reduction Rule 7.2.Set Y := Y \ N G (I). Reduction Rule 7.3.If |Y | ≥ 2, then delete |Y | − 1 arbitrary vertices of Y and set I := I ∪ Y .Claim 7.3.Reduction Rules 7.2 and 7.3 are safe and the set I constructed by Reduction Rule 7.3 is independent.
Clearly, |S | = |S| ≥ .Assume now that v ∈ Y \ N G (I).Note that the vertices of Y \ N G (I) are true twins in G, that is, for every x, y ∈ Y \ N G (I), N G [x] = N G [y].Hence, we can assume without loss of generality that u was not deleted by Reduction Rule 7.3 and S ⊆ V (G ).

Claim 7 . 4 .
Reduction Rule 7.4 is safe.Proof of Claim 7.4.Denote by G the graph obtained from G by the application of Reduction Rule 7.4 for a vertex u ∈ X.It is sufficient to show that if G has an independent set S with |S| ≥ , then G has an independent set of size at least .If u / ∈ S, then S is an independent set of G .Assume that u ∈ S. Consider the set I. Observe that |S ∩ I| ≤ |I| − 2k.By Claim 7.2 (v), |S ∩ X| ≤ 2k.We obtain that |I| ≥ |S|.Since I is an independent set in G , the claim follows.
Chordal − ke and construct the corresponding triangulation H if this holds.Let H