FixedParameter Tractable Distances to Sparse Graph Classes
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Abstract
We show that for various classes \(\mathcal {C}\) of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph G to \(\mathcal {C}\) is fixedparameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise firstorder definable is \(\mathrm {FPT} \). The second shows that determining the elimination distance of a graph G to a minorclosed class \(\mathcal {C}\) is \(\mathrm {FPT} \). We demonstrate that several prior results (of Golovach, Moser and Thilikos and Mathieson) on the fixedparameter tractability of distance measures are special cases of our first method.
Keywords
Fixedparameter tractable Parameterized complexity Graph theory Sparse graphs Nowhere dense Excluded minor Minorclosed Deletion distance Elimination distance Distance1 Introduction
The study of parameterized algorithmics for graph problems has thrown up a large variety of structural parameters of graphs. Among these are parameters that measure the distance of a graph G to a class \(\mathcal {C}\) in some way. The simplest such measures are those that count the number of vertices or edges that one must delete (or add) to G to obtain a graph in \(\mathcal {C}\). A common motivation for studying such parameters is that if a problem one wishes to solve is tractable on the class \(\mathcal {C}\), then the distance to \(\mathcal {C}\) provides an interesting parameterization of that problem (called distance to triviality by Guo et al. [19]). Other examples of this include the study of modulators to graphs of bounded treewidth in the context of kernelization (see [13, 15]) or the parameterizations of colouring problems (see [22]). On the other hand, determining the distance of an input graph G to a class \(\mathcal {C}\) is, in general, a computationally challenging problem in its own right. Such problems have also been extensively studied with a view to establishing their complexity when parameterized by the distance. A canonical example is the problem of determining the size of a minimum vertex cover in a graph G, which is just the vertexdeletion distance of G to the class of edgeless graphs. More generally, Cai [4] studies the parameterized complexity of distance measures defined in terms of addition and deletion of vertices and edges to hereditary classes \(\mathcal {C}\). Counting deletions of vertices and edges gives a rather simple notion of distance, and many more involved notions have also been studied. Classic examples include the crossing number of a graph which provides one notion of distance to the class of planar graphs or the treewidth of a graph which can be seen as a measure of distance to the class of trees, as argued in [19]. Another recently introduced measure is elimination distance, defined in [3] where it was shown that graph isomorphism is \(\mathrm {FPT} \) when parameterized by elimination distance to a class of graphs of bounded degree.
In this paper we consider the fixedparameter tractability of a variety of different notions of distance to various different classes \(\mathcal {C}\) of sparse graphs. We establish two quite general techniques for establishing that such a distance measure is \(\mathrm {FPT} \). The first builds on the recent result of Grohe et al. [18] which shows that the problem of evaluating firstorder formulas on any nowhere dense class of graphs is \(\mathrm {FPT} \) with the formula as parameter. We extract from their proof of this result a general statement about the fixedparameter tractability of definable sparse classes. To be precise, we show that parameterized problems that are both slicewise nowhere dense and slicewise firstorder definable (these terms are defined precisely below) are \(\mathrm {FPT} \). As an application of this, it follows that if \(\mathcal {C}\) is a nowhere dense class of graphs that is definable by a firstorder formula, then the parameterized problem of determining the distance of a graph G to \(\mathcal {C}\) is \(\mathrm {FPT} \), for various notions of distance that can be themselves so defined. In particular, we get that various forms of edit distance to classes of degreeconstrained graphs are \(\mathrm {FPT} \). We illustrate the power of this method by showing that it includes as special cases prior results by Golovach [16], Moser and Thilikos [26] and Mathieson [23, 24] obtained by more specific methods. Another interesting application is obtained by considering elimination distance of a graph G to the class \(\mathcal {C}\) of empty graphs. This is nothing other than the treedepth of G. While elimination distance to a class \(\mathcal {C}\) may not, in general, be firstorder definable, it is in the particular case where \(\mathcal {C}\) is the class of empty graphs. Thus, we obtain as an application of our method the result that treedepth is \(\mathrm {FPT} \), a result previously known from other algorithmic meta theorems (see [27, Theorem 17.2]). The method of establishing that a parameterized problem is \(\mathrm {FPT} \) by establishing that it is slicewise nowhere dense and slicewise firstorder definable appears to be a powerful method of some generality which should find application beyond these examples.
Our second general method specifically concerns elimination distance to a minorclosed class \(\mathcal {C}\). We show that this measure is fixedparameter tractable for any such \(\mathcal {C}\), answering an open question posed in [3]. Note that while a proper minorclosed class is always nowhere dense, it is not generally firstorder definable (for instance, neither the class of acyclic graphs nor the class of planar graphs is), and elimination distance to such a class is also not known to be firstorder definable. Thus, our results on the tractability of slicewise firstorder definable classes do not apply here. Instead, we build on work of Adler et al. [1] to show that from a finite list of the forbidden minors characterising \(\mathcal {C}\), we can compute the set of forbidden minors characterising the graphs at elimination distance k to \(\mathcal {C}\). Adler et al. show how to do this for apex graphs, from which one immediately obtains the result for graphs that are k deletions away from \(\mathcal {C}\). To extend this to elimination distance k, we show how we can construct the forbidden minors for the closure of a minorclosed class under disjoint unions.
In Sect. 2 we present the definitions necessary for the rest of the paper. Section 3 establishes our result for slicewise firstorder definable and slicewise nowhere dense problems and Sect. 4 gives some applications of the general method. Section 5 establishes that the problem of determining elimination distance to any minorclosed class is \(\mathrm {FPT} \). Some open questions are discussed in Sect. 6.
2 Preliminaries
FirstOrder Logic We assume some familiarity with firstorder logic for Sect. 3. A (relational) signature \(\sigma \) is a finite set of relation symbols, each with an associated arity. A \(\sigma \) structure A consists of a set V(A) and for each kary relation symbol \(R \in \sigma \) a relation \(R(A) \subseteq V(A)^k\). Our structures will mostly be (coloured) graphs, so \(\sigma = \{E\}\) or \(\sigma = \{E, C_1, C_2, \ldots , C_r\}\) where E is binary and the \(C_i\) are unary relation symbols. A graph G is then a \(\sigma \)structure with vertex set V(G), edge relation E(G), and colours \(C_i(G)\).
We sometimes need to define formulas of firstorder logic by relativisation, and we define the notion here.
Definition 1

for atomic \(\varphi , \varphi ^{[x.\psi ]}\) is the same as \(\varphi \);

\((\varphi _1 \vee \varphi _2)^{[x.\psi ]}\) is \(\varphi _1^{[x.\psi ]} \vee \varphi _2^{[x.\psi ]}\);

\((\lnot \varphi )^{[x.\psi ]}\) is \(\lnot (\varphi )^{[x.\psi ]}\); and

\((\exists v \, \varphi ')^{[x.\psi ]}\) is \(\exists v (\psi [v/x] \wedge (\varphi ')^{[x.\psi ]})\). Here \(\psi [v/x]\) denotes the result of replacing the free occurrences of x in \(\psi \) with v in a suitable way avoiding capture.
The key idea here is that \(\varphi ^{[x.\psi ]}\) is true in a graph G if, and only if, \(\varphi \) is true in the subgraph of G induced by the vertices that satisfy \(\psi (x)\). In particular the relatvisation of \(\forall v \, \varphi '\) is \(\forall v (\psi [v/x] \rightarrow (\varphi ')^{[x.\psi ]})\).
Note that the variable x that is free in \(\psi \) is bound in \(\varphi ^{[x.\psi ]}\). Other variables that appear free in \(\psi \) remain free in \(\varphi ^{[x.\psi ]}\). We stress this as it is needed in Proposition 2 where nested relativisations are used.
Parameterized Complexity Parameterized complexity theory is a twodimensional approach to the study of the complexity of computational problems. We find it convenient to define problems as classes of structures rather than strings, following the textbook of Flum and Grohe [12]. A problem \(Q \subseteq \mathop {{ str}}(\sigma )\) is an (isomorphismclosed) class of \(\sigma \)structures given some signature \(\sigma \). A parameterization is a computable function \(\kappa \,{:}\,\mathop {{ str}}(\sigma ) \rightarrow \mathbb {N}\). We say that Q is fixedparameter tractable with respect to \(\kappa \) if we can decide whether an input \(A \in \mathop {{ str}}(\sigma )\) is in Q in time \(O(f(\kappa (A)) \cdot A^c)\), where c is a constant and f is some computable function. For a thorough discussion of the subject we refer to the books by Downey and Fellows [9], Flum and Grohe [12] and Niedermeier [29].
A parameterized problem \((Q, \kappa )\) is slicewise firstorder definable if there is a computable function \(f : \mathbb {N}\rightarrow \) FO\([\sigma ]\) such that a \(\sigma \)structure A with \(\kappa (A) \le i\) is in Q if, and only if, \(A \models f(i)\). Slicewise definability of problems in a logic was introduced by Flum and Grohe [11].
Graph Theory A graph G is a set of vertices V(G) and a set of edges \(E(G)\subseteq V(G) \times V(G)\). We assume that graphs are loopfree and undirected, i.e. that E is irreflexive and symmetric. We mostly follow the notation in Diestel [8]. For a set \(S \subseteq V(G)\) of vertices, we write \(G {\setminus } S\) to denote the subgraph of G induced by \(V(G){\setminus } S\).
Let \(r \in \mathbb {N}\). An rindependent set in a graph G is a set of vertices of G such that their pairwise distance is at least r.
A graph H is a minor of a graph G, written Open image in new window , if there is a map, called the minor map, that takes each vertex \(v \in V(H)\) to a tree \(T_v\) that is a subgraph of G such that for any \(u \ne v\) the trees are disjoint, i.e. \(T_v \cap T_u = \emptyset \), and such that for every edge \(uv \in E(H)\) there are vertices \(u' \in T_u, v' \in T_v\) with \(u'v' \in E(G)\). A class of graphs \(\mathcal {C}\) is minorclosed if Open image in new window and \(G \in \mathcal {C}\) implies \(H \in \mathcal {C}\).
The set of minimal excluded minors \(M(\mathcal {C})\) is the set of graphs in the complement of \(\mathcal {C}\) such that for each \(G \in M(\mathcal {C})\) all proper minors of G are in \(\mathcal {C}\). By the Robertson–Seymour Theorem [30] the set \(M(\mathcal {C})\) is finite for every minorclosed class \(\mathcal {C}\). It is a consequence of this theorem that membership in a minorclosed class can be tested in \(O(n^3)\) time. For a set M of graphs, we write \(\mathrm {Forb}(M)\) for the class of graphs which forbid M as minors, i.e. Open image in new window .
Let \(r \in \mathbb {N}\). A minor H of G is a depthr minor of G, written Open image in new window , if there is a minor map that takes vertices in H to trees that have radius at most r. A class of graphs \(\mathcal {C}\) is nowhere dense if for every \(r \in \mathbb {N}\) there is a graph \(H_r\) such that for no \(G \in \mathcal {C}\) we have Open image in new window . A nowheredense class of graphs \(\mathcal {C}\) is called effectively nowhere dense if there is a computable function h from integers to graphs such that if \(G \in \mathcal {C}\) then for all r we have Open image in new window . We are only interested in effectively nowheredense classes so we simply use the term nowhere dense to mean effectively nowhere dense.
We say that a parameterized graph problem \((Q, \kappa )\) is slicewise nowhere dense if there is a computable function h from pairs of integers to graphs such that for all \(i \in \mathbb {N}\), we have if \(G \in Q\) and \(\kappa (G) \le i \) then for all r we have Open image in new window . We call h the parameter function of Q.
For a class of graphs \(\mathcal {C}\) we denote the closure of \(\mathcal {C}\) under taking disjoint unions by \(\overline{\mathcal {C}}\). We say that a graph G is an apex graph over a class \(\mathcal {C}\) of graphs if there is a vertex \(v \in V(G)\) such that the graph \(G {\setminus } \{v\} \in \mathcal {C}\). The class of all apex graphs over \(\mathcal {C}\) is denoted \({\mathcal {C}}^{\text {apex}}\).
A graph G has deletion distance k to a class \(\mathcal {C}\) if there are k vertices \(v_1, \dots , v_k \in V(G)\) such that \(G {\setminus } \{v_1, \dots , v_k\} \in \mathcal {C}\).
A richer notion of distance to a graph class \(\mathcal {C}\), inspired by the definition of treedepth, is introduced in [3], and we define it next.
3 A General Method for Editing Distances
In this section we establish a general technique for showing that certain definable parameterized problems on graphs are \(\mathrm {FPT} \). As an application, we show that certain natural distance measures to sparse graph classes are \(\mathrm {FPT} \). To be precise, we show that if a parameterized problem is both slicewise firstorder definable and slicewise nowhere dense, then it is \(\mathrm {FPT} \). In particular, this implies that if we have a class \(\mathcal {C}\) that is firstorder definable and nowhere dense and the distance measure we are interested in is also firstorder definable (that is to say, for each k there is a formula that defines the graphs of distance k from \(\mathcal {C}\)), then the problem of determining the distance is \(\mathrm {FPT} \). More generally, if we have a parameterized problem \((Q,\kappa )\) that is slicewise nowhere dense and slicewise firstorder definable, and a measure of distance to it is definable in the sense that for any values of k and d, there is a firstorder formula defining the graphs of distance d to the class \(\{G \mid G \in Q \text{ and } \kappa (G) \le k\}\), then the problem of deciding whether a graph has distance at most d to this class is \(\mathrm {FPT} \) parameterized by \(d+k\). In Sect. 4 we show that this provides a unifying account of a number of existing results in the literature by giving a single method of proof for them.
The method is an adaptation of the main algorithm in Grohe et al. [18]. Since the proof of our results is essentially a modification of the central construction in [18], rather than give a full account, we state the main results they prove and explain briefly how the proofs can be adapted for our purposes. For a full proof, this section is best read in conjunction with the paper [18]. In Sect. 3.1 we give an overview of the key elements of the construction from [18] and state all the definitions we require to formulate our results. Section 3.2 then gives our main result and Sect. 4 derives some consequences for distance measures.
3.1 Evaluating Formulas on Nowhere Dense Classes
The key result of [18] is:
Theorem 1
[18, Theorem 1.1] For every nowhere dense class \(\mathcal {C}\) and every \(\epsilon > 0\), every property of graphs definable in firstorder logic can be decided in time \(O(n^{1+\epsilon })\) on \(\mathcal {C}\).
The proof of this theorem rests, in turn, on two others, stated as Theorems 2 and 3 below. In order to formally state those results and explain how they can be adapted to our purposes, we need to formulate some definitions.
To evaluate a basic local sentence such as (1) in a graph G, it suffices to determine for each vertex v whether the neighbourhood of radius d around it satisfies \(\theta \) and then check whether in set of vertices for which this is true there is a 2dindependent set. If the neighbourhoods in G are structurally simpler than G, this may provide an efficient means of evaluating a firstorder sentence. For instance, in a class \(\mathcal {C}\) of bounded local treewidth, the dneighbourhoods of vertices have bounded treewidth, and if \(\mathcal {C}\) has locally excluded minors then for every d, the dneighbourhoods exclude some graph as a minor. If \(\mathcal {C}\) is a nowheredense class, we do not have a structurally simpler characterisation of the dneighbourhoods occurring in graphs in \(\mathcal {C}\). Instead we rely on the property that such classes are quasiwide. This is a notion of sparseness for graphs introduced in [5, 6] (for the connection to nowheredense classes, see [27]). A graph class \(\mathcal {C}\) is said to be quasiwide if there is a function s such that for any d and N and any graph \(G \in \mathcal {C}\), if A is a sufficiently large set of more vertices in G, then G contains a bottleneck set S of at most s(d) vertices such that in \(G{\setminus } S, A\) contains a dindependent set of size N.
This suggests strategy for evaluating a formula \(\varphi \) in a graph G that comes from a class \(\mathcal {C}\) that is nowhere dense and closed under taking subgraphs. We identify a bottleneck set S and remove these vertices from the graph, colouring each remaining vertex v according to which elements of S are neighbours of v. It is easy to translate the formula \(\varphi \) into a formula \(\varphi '\) in a vocabulary expanded with these colours such that \(\varphi '\) is true in the coloured graph \(G{\setminus } S\) if, and only if, \(\varphi \) is true in G. The existence of a large set of vertices in \(G{\setminus } S\) that are pairwise far apart means that we can evaluate local sentences in neighbourhoods around these vertices in (total) amortized time dependent on the size of the graph. This evaluation is done recursively, since the neighbourhood of a vertex v is also in the class \(\mathcal {C}\) (by the assumption of closure under taking subgraphs). The difficulty with this approach is that, if all we know about the dneighbourhood of a vertex is that it is also in \(\mathcal {C}\), we cannot bound the depth of recursion by a constant. Thus the vocabulary of the formulas \(\varphi '\) we construct, and hence also their size, is no longer dependent solely on parameters. Grohe et al. circumvent this difficulty through two innovative methods. The first is to define a colouring based on sparse neighbourhood covers and show that such covers exist in nowheredense classes of graphs. The second is an alternative way of amortizing the quantifier rank of the recursively defined formulas in the expanded vocabularies, defining a discounted measure of rank. We expand on these notions next.
For \(r \in \mathbb {N}\), an r neighbourhood cover of a graph G is a set \(\mathcal {X}\) of connected subgraphs of G such that for every \(v \in V(G)\) there is an \(X \in \mathcal {X}\) that contains the rneighbourhood of v. The elements of \(\mathcal {X}\) are called clusters. The radius of \(\mathcal {X}\) is the maximum radius of any of its clusters. The degree of a vertex v is the number of clusters that contain v an the maximum degree of \(\mathcal {X}\) is the maximum over all \(v \in V(G)\) of the degree of v. These definitions allow us to state the following theorem.
Theorem 2
[18, Theorem 6.2] Let \(\mathcal {C}\) be a nowhere dense class of graphs. There is a function f such that for all \(r \in \mathbb {N}\) and \(\epsilon > 0\) and all graphs \(G \in \mathcal {C}\) with \(n \ge f(r, \epsilon )\) vertices, there exists an rneighbourhood cover of radius at most 2r and maximum degree at most \(n^\epsilon \) and this cover can be computed in time \(f(r, \epsilon ) \cdot n^{1+ \epsilon }\). Furthermore, if \(\mathcal {C}\) is effectively nowhere dense, then f is computable.
In this theorem, f is a function of r and \(\epsilon \) and depends on the class \(\mathcal {C}\) in the sense that it is determined, for an effectively nowhere dense \(\mathcal {C}\) by its parameter function. While the algorithm of [18] assumes that the input graph G comes from the class \(\mathcal {C}\), we can say something more. For a fixed nowhere dense class \(\mathcal {C}\), where we know the parameter function h, we can, given G, r and \(\epsilon \), compute a bound on the running time of the algorithm from Theorem 2. By running the algorithm to this bound, we have the following as a direct consequence of the proof of Theorem 2.
Lemma 1
Let \(\mathcal {C}\) be a nowhere dense class of graphs. There is a function f such that for all \(r \in \mathbb {N}\) and \(\epsilon > 0\) and all graphs \(G \in \mathcal {C}\) with \(n \ge f(r, \epsilon )\) vertices, there exists an rneighbourhood cover of radius at most 2r and maximum degree at most \(n^\epsilon \). There is an algorithm that given an arbitrary graph G runs in time \(f(r, \epsilon ) \cdot n^{1+ \epsilon }\) and that computes this cover or determines that \(G \not \in \mathcal {C}\). Furthermore, if \(\mathcal {C}\) is effectively nowhere dense, then f is computable.
The logic \(\textsf {FO}^+\) is defined by extending \(\textsf {FO}\) with an atomic formula \(\mathrm {dist}_d(x,y)\) for each \(d\in \mathbb {N}\) and each pair of variables x, y. The idea is that in this expanded logic, the assertion that x and y are at distance at least d does not require any quantifiers. However, we lose a key feature of firstorder logic that is essential to parameterized algorithms and that is that, for each q, there are, up to equivalence, only finitely many distinct sentences of \(\textsf {FO}\) of quantifierrank q. To recover this property for \(\textsf {FO}^+\), Grohe et al. define a discounted quantifier rank measure as follows. We say that \(\varphi \) of \(\textsf {FO}^+\) has qrank m if it has quantifierrank at most m and for any atomic subformula \(\mathrm {dist}_d(x,y)\) which occurs within the scope of i quantifiers we have \(d \le (4q)^{q+mi}\). It can then be shown that there are for each k,q and m, up to equivalence, only finitely many distinct formulas of \(\textsf {FO}^+\) with qrank m in the free variables \(x_1,\ldots ,x_k\). Fix \(\varphi ^+(\sigma ,k,q,m)\) a finite set of representative formulas in the vocabulary \(\sigma \), including one of each class up to logical equivalence. We then define the vocabulary \(\sigma \star q\) as the expansion of \(\sigma \) with a unary relation \(P_{\varphi }\) for each \(\varphi \in \varphi ^+(\sigma ,1,q,q)\).
Theorem 3
An important tool for constructing \(G \star _{\mathcal {X}}^{q+1} q\) is a game characterisation of nowhere dense classes. The game has three parameters: \(\ell , m, r\). In the \((\ell , m, r)\) Splitter game two players Connector and Splitter play against each other. In each round Connector chooses a vertex u, and Splitter has to respond with a set A of vertices of size at most m in the rneighbourhood of u. In the next round the graph is the neighbourhood of u with the vertices from A removed. If the graph is empty, Splitter wins. If Connector survives for more than \(\ell \) rounds, she wins. Grohe et al. [18, Theorem 4.2] prove that if \(\mathcal {C}\) is a nowhere dense class, then there are \(\ell , m\) such that Splitter has a winning strategy on the \((\ell , m, 2r)\) Splitter game on every graph in \(\mathcal {C}\).
The Splitter’s strategy on a graph G (which can be efficiently computed) is the essential tool in the construction of \(G \star _{\mathcal {X}}^{q+1} q\). The inductive procedure used to compute \(G \star _{\mathcal {X}}^{q+1} q\) from G is outlined in [18, Proof of Theorem 8.1]. We note that the termination of the algorithm depends on the length of the game—which is bounded by a constant since \(\mathcal {C}\) is nowhere dense. The strategy to compute Splitter’s moves is described in [18, Remark 4.3]. Since the run time of the algorithm to compute \(G \star _{\mathcal {X}}^{q+1} q\) only depends on q and the length of the Splitter game and we can compute this in advance, we can once again extract the fact that if we start with an arbitrary graph G, we can efficiently either transform it into \(G \star _{\mathcal {X}}^{q+1} q\) or determine that it is not in the class \(\mathcal {C}\). This is summed up in the following lemma.
Lemma 2
Let \(\mathcal {C}\) be a nowhere dense class of graphs. For every \(\epsilon > 0\) there is an algorithm that runs in time \(O(f(q) \cdot n^{1+\epsilon })\) for some function f, and which given a graph G returns \(G \star _{\mathcal {X}}^{q+1} q\) or determines that \(G \not \in \mathcal {C}\).
The problem is shown to be \(\mathrm {FPT} \) on nowhere dense classes of graphs [18, Theorem 5.1]. Since the runtime of the algorithm depends on the length of the Splitter game and Splitter’s strategy, and this can be bounded in advance, [18, Theorem 5.1] can be restated as follows:
Lemma 3
Let \(\mathcal {C}\) be a nowhere dense class of graphs. Then there is an algorithm and a function f such that for every \(\epsilon > 0\) the algorithm runs in time \(f(\epsilon , r, k)\) and either solves the Distance Independent Set problem or determines that \(G \not \in \mathcal {C}\). Furthermore, if \(\mathcal {C}\) is effectively nowhere dense, then f is computable.
This is all we need to evaluate \(\hat{\phi }\) on \(G \star _{\mathcal {X}}^{q+1} q\), which is equivalent to evaluating \(\phi \) on G by Theorem 3.
3.2 Deciding Definable Nowhere Dense Problems
The main result of [18] establishes that checking whether \(G\models \phi \) is \(\mathrm {FPT} \) when parameterized by \(\phi \) provided that G comes from a known nowhere dense class \(\mathcal {C}\). Thus, the formula is arbitrary, but the graphs come from a restricted class. In Sect. 3.1 above we give an account of this proof from which we can extract the observation that the algorithm can be modified to work for an arbitrary input graph G with the requirement that the algorithm may simply reject the input if G is not in \(\mathcal {C}\). This suggests a tractable way of deciding \(G\models \phi \) provided that \(\phi \) defines a nowhere dense class. Now the graph is arbitrary, but the formula comes from a restricted class. We formalise the result in the following theorem:
Theorem 4
Let \((Q, \kappa )\) be a problem that is slicewise firstorder definable and slicewise nowhere dense. Then \((Q, \kappa )\) is fixedparameter tractable.
Proof
 Step 1:

Compute \(\varphi \) and the parameter function. Since \((Q, \kappa )\) is slicewise firstorder definable, we can compute from i a firstorder formula \(\phi \) which defines the class of graphs \(C_i = \{H \mid H \in Q \text{ and } \kappa (H) \le i\}\). Moreover, since \((Q, \kappa )\) is slicewise nowhere dense, we can compute from i an algorithm that computes the parameter function h for \(C_i\).
 Step 2:

Obtain \(\hat{\varphi }\) from \(\varphi \). By the RankPreserving Locality Theorem (Theorem 3), we can compute from \(\varphi \) the formula \(\hat{\varphi }\) and a radius r.
 Step 3:

Find a small cover \(\mathcal {X}\) for G. By Lemma 1, we can either find a cover \(\mathcal {X}\) for G, or reject if the algorithm determine that \(G \not \in C_i\).
 Step 4:

Simulate Splitter game to compute \(G'\). By Lemma 2 we obtain \(G'\) or reject if the algorithm determines that \(G \not \in C_i\).
 Step 5:

Evaluate \(\hat{\varphi }\) on \(G'\). Finally to evaluate \(\hat{\varphi }\) on \(G'\), we need to solve the distance independent set problem. We can do this by Lemma 3. Since evaluating \(\hat{\varphi }\) on \(G'\) is equivalent to evaluating \(\varphi \) on G this allows us to decide whether \(G \in Q\).
4 Applications
In this section we illustrate the power of Theorem 4 by showing that a number of known fixedparameter tractability results can be obtained as direct consequences of the theorem. In Sect. 4.1, we start with the simple observation that if a parameterized problem is slicewise firstorder definable than so is deletion distance to the problem (suitably parameterized). Examples of this include previous results of Moser and Thilikos [26]. We consider more general edit distances in Sect. 4.2 and show that of Mathieson [23, 25] can be obtained as special cases of our result. Finally, in Sect. 4.3, we consider the problem of computing the treedepth of a graph. Again, this problem is known to be FPT, and the novelty here is in constructing the firstorder definitions that show it is slicewise firstorder definable. The consideration of treedepth also leads naturally to considering, more generally, elimination distance to sparse classes and this topic is taken up in Sect. 5.
4.1 Deletion Distance
Proposition 1
If \((Q,\kappa )\) is slicewise nowhere dense and slicewise firstorder definable then Deletion Distance to Q is \(\mathrm {FPT} \).
Proof
To see that Deletion Distance to Q is also slicewise nowhere dense, let h be the parameter function for Q. If the graph h(i, r) has m vertices, then \(K_m\) is not a depthrminor of any graph in \(\mathcal {C}_i\). Then a graph which has deletion distance k to \(\mathcal {C}_i\) cannot have \(K_{m+k}\) as a depthrminor. Indeed, suppose Open image in new window and \(G{\setminus } S \in \mathcal {C}_i\) where S is a set of k vertices. Vertices from S can appear in the images of at most k vertices from \(K_{m+k}\) under the minor map. Thus, this minor map also witnesses that Open image in new window , a contradiction.\(\square \)
Consider as an example deletion distance k to maximum degree d.
Example 1
Moser and Thilikos [26] showed that deleting k vertices to obtain a dregular graph is fixedparameter tractable parameterized by \(k + d\). Since the class of dregular graphs is also firstorder definable and nowhere dense for any d, their result is also a consequence of Theorem 4.
4.2 Edit Distances to Graph Classes Defined by Degree Constraints
Instead of deletion distance (defined by deleting vertices), we can also consider more general graph editing distances (defined through more general edit operations on the graph), e.g. modifying the graph by adding or deleting edges.
Thus, an analogue of Proposition 1 can be obtained for any edit distance where the allowed edit operations are a combination of vertex and edge deletions and additions. In the following we discuss this in more detail, where the class we are editing to is defined by degree constraints.
Mathieson [25] shows that the problem is fixedparameter tractable for any S and parameter \(k + d\). Inspired by Stewart [31], Mathieson shows that the problem is firstorder definable (with the size of the formula depending on k and d), by considering the incidence graph as a relational structure. (For the weighted version of the problem, he adds a unary relation for every possible weight.) Since a graph that can be edited to be regular must its degree bounded by \(k + d\) it is therefore also nowhere dense. Thus the result also follows directly from Theorem 4.
Building on this, Golovach [16] gives a concrete algorithm that edits a graph so that every vertex has a given degree at most d using at most k edge additions/deletions.
More recently, Mathieson [24] looks at more general versions of degree constraint problems. He considers three notions of regularity: edgedegreeregular, edgeregular and stronglyregular. He studies the problems of editing to these three notions of regularity.
The edgedegree of an edge uv is the sum of the degrees of the endpoints of \(d(u) + d(v)\) and a graph is edgedegreeregular if all edges uv have the same edgedegree.
The two other notions combine the degrees of vertices and common neighbourhoods of endpoints of edges (and nonedges). A graph is \((r,\lambda )\) edgeregular if every vertex has degree r and every edge uv has \(N(u) \cap N(v) = \lambda \). A graph is \((r, \lambda , \mu )\) stronglyregular if it is \((r, \lambda )\)edgeregular and for every pair u, v of nonadjacent vertices we have \(N(u) \cap N(v) = \mu \). This is the standard notion of a strongly regular graph as introduced by Bose [2].
4.3 TreeDepth
Recall that treedepth is a graph parameter that lies between the widely studied parameters vertex cover number and tree width. It has interesting connections to nowhere dense graph classes, and can itself be interpreted as a distance measure (elimination distance to the empty graph). For convenience we give the usual definition here:
Definition 2
Note that a graph has treedepth k if and only if it has elimination distance k to the class of empty graphs. So one can think of elimination distance as a natural generalisation of treedepth.
It is known that the problem of determining the treedepth of graph is \(\mathrm {FPT} \), with treedepth as the parameter (see [28, Theorem 7.2]). We now give an alternative proof of this, using Theorem 4. It is clear that for any k, the class of graphs of treedepth at most k is nowhere dense. We show below that it is also firstorder definable.
Proposition 2
For each \(k \in \mathbb {N}\) there is a firstorder formula \(\varphi _k\) such that a graph G has treedepth k if and only if \(G \models \varphi _k\).
Proof
We use the fact that in a graph of treedepth less than k, there are no paths of length greater than \(2^k\) [28, Section 6.2]. This allows us, in the inductive definition of treedepth above, to replace the condition of connectedness (which is not firstorder definable) with a firstorder definable condition on vertices at distance at most \(2^k\).
Recall that \(\mathrm {dist}_d(u,v)\) is the firstorder formula with free variables u and v that is satisfied by a pair of vertices in a graph G if, and only if, they have distance at most d in G. Note that the formula \(\mathrm {dist}_d^{[x.x \ne w]}(u,v)\) is then a formula with three free variables u, v, w which defines those u, v which have a path of length d in the graph obtained by deleting the vertex w.
We can now define the formula \(\phi _k\) by induction. Only the empty graph has treedepth 0, so \(\phi _0 := \lnot \exists v(v = v)\).
While the proof of Proposition 1 shows that deletion distance to any slicewise firstorder definable class is also slicewise firstorder definable, Proposition 2 shows that elimination distance to the particular class of empty graphs is slicewise firstorder definable. It does not establish this more generally for elimination distance to any slicewise nowhere dense class—that remains an open question. We conjecture that elimination distance to a slicewise nowhere dense class is not firstorder definable.
5 Elimination Distance to Classes Characterised by Excluded Minors
It is not difficult to show that the class of graphs which have elimination distance k to a minorclosed class \(\mathcal {C}\) is also a minorclosed class. Indeed, this can be seen directly from an alternative characterisation of elimination distance that we establish below. The characterisation is in terms of the iterated closure of \(\mathcal {C}\) under the operation of disjoint unions and taking the class of apex graphs.We introduce a piece of notation for this in the next definition. Recall that we write \({\mathcal {C}}^{\text {apex}}\) for the class of all the apex graphs over \(\mathcal {C}\), and that we write \(\overline{\mathcal {C}}\) for the closure of \(\mathcal {C}\) under disjoint unions.
Definition 3
For a class of graphs \(\mathcal {C}\), let \(\mathcal {C}_0 := \mathcal {C}\), and \(\mathcal {C}_{i+1} := \overline{{\mathcal {C}_i}^{\text {apex}}}\).
We show next that the class \(\mathcal {C}_k\) is exactly the class of graphs at elimination distance k from \(\mathcal {C}\).
Proposition 3
Let \(\mathcal {C}\) be a class of graphs and \(k \ge 0\). Then \(\mathcal {C}_k\) is the class of all graphs with elimination distance at most k to \(\mathcal {C}\).
Proof
We prove this by induction. Only the graphs in \(\mathcal {C}\) have elimination distance 0 to \(\mathcal {C}\), so the statement holds for \(k = 0\).
Suppose the statement holds for k. If \(G \in \mathcal {C}_{k+1}\), then G is a disjoint union of graphs \(G_1, \dots , G_s\) from \({\mathcal {C}_{k}}^{\text {apex}}\), so we can remove at most one vertex from each of the \(G_i\) and obtain a graph in \(\mathcal {C}_{k}\). Thus the elimination distance of G to \(\mathcal {C}_{k}\) is 1, and by induction the elimination distance to \(\mathcal {C}\) is \(k+1\). Conversely, if G has elimination distance \(k+1\) to \(\mathcal {C}\), then we can remove a vertex from each component of G to obtain a graph \(G'\) with elimination distance k to \(\mathcal {C}\). Using the induction hypothesis each component of \(G'\) is in \(\mathcal {C}_{k}\), and thus \(G \in \mathcal {C}_{k+1}\).\(\square \)
It is easy to see that if \(\mathcal {C}\) is a minorclosed class of graphs then so is \(\mathcal {C}_k\) for any k. Indeed, it is wellknown that \({C}^{\text {apex}}\) is minorclosed for any minorclosed \(\mathcal {C}\), so we just need to note that \(\overline{\mathcal {C}}\) is also minorclosed. But it is clear that if H is a minor of a graph G that is the disjoint union of graphs \(G_1,\ldots ,G_s\), then H itself is the disjoint union of minors of \(G_1,\ldots ,G_s\). Thus, the class of graphs of elimination distance at most k to a minorclosed class \(\mathcal {C}\) is itself minorclosed. We next show that we can construct the set of its minimal excluded minors from the corresponding set for \(\mathcal {C}\).
To obtain \(M(\mathcal {C}_k)\), we need to iteratively compute \(M({\mathcal {C}}^{\text {apex}})\) and \(M(\overline{\mathcal {C}})\) from \(M(\mathcal {C})\). Adler et al. [1] show that from the set of minimal excluded minors \(M(\mathcal {C})\) of a class \(\mathcal {C}\), we can compute \(M({\mathcal {C}}^{\text {apex}})\):
Theorem 5
[1, Theorem 5.1] There is a computable function that takes the set of graphs \(M(\mathcal {C})\) characterising a minorclosed class \(\mathcal {C}\) to the set \(M({\mathcal {C}}^{\text {apex}})\).
We next aim to show that from \(M(\mathcal {C})\) we can also compute \(M(\overline{\mathcal {C}})\). Together with Theorem 5 this implies that from \(M(\mathcal {C})\) we can compute \(M(\mathcal {C}_k)\).
We begin by characterising minorclosed classes that are closed under disjoint unions in terms of the connectedness of their excluded minors.
Lemma 4
Let \(\mathcal {C}\) be a class of graphs closed under taking minors. Then \(\mathcal {C}\) is closed under taking disjoint unions iff each graph in \(M(\mathcal {C})\) is connected.
Proof
Let \(\mathcal {C}\) be a minorclosed class of graphs, and let \(M(\mathcal {C}) = \{H_1, \dots , H_s\}\) be its set of minimal excluded minors.
Suppose each of the graphs in \(M(\mathcal {C})\) is connected. Let \(H \in M(\mathcal {C})\) and let \(G = G_1 \oplus \dots \oplus G_r\) be the disjoint union of graphs \(G_1, \dots , G_r \in \mathcal {C}\). Because H is connected, we have that Open image in new window if and only if Open image in new window for one \(1 \le i \le r\). So, since all the \(G_i \in \mathcal {C}\), we have Open image in new window and thus \(G \in \mathcal {C}\). This shows that \(\mathcal {C}\) is closed under taking disjoint unions.
Conversely, assume one of the graphs \(H \in M(\mathcal {C})\) is not connected, and let \(A_1, \dots , A_t\) be its connected components. Then \(A_1, \dots , A_t \in \mathcal {C}\), since each \(A_i\) is a proper minor of H, and H is minorminimal in the complement of \(\mathcal {C}\). However, \(A_1 \oplus \dots \oplus A_t = H \not \in \mathcal {C}\).\(\square \)
Definition 4
For a graph G with connected components \(G_1, \dots , G_r\), let \(\mathcal {H}\) denote the set of connected graphs H with \(V(H) = V(G)\) and such that the subgraph of H induced by \(V(G_i)\) is exactly \(G_i\). We define the connection closure of G to be the set of all minimal (under the subgraph relation) graphs in \(\mathcal {H}\). The connection closure of a set of graphs is the union of the connection closures of the graphs in the set.
Note that if G has e edges and m components, then any graph in the connection closure of G has exactly \(e+m1\) edges. This is because it has G as a subgraph and in addition \(m1\) edges corresponding to a tree on m vertices connecting the m components.
Lemma 5
Let \(\mathcal {C}\) be a minorclosed class of graphs. Then \(M(\overline{\mathcal {C}})\) is the set of minorminimal graphs in the connection closure of \(M(\mathcal {C})\).
Proof
Let \(\mathcal {C}\) be a minorclosed class of graphs, with \(M(\mathcal {C})\) its set of minimal excluded minors, and let \(\hat{M}\) be the connection closure of \(M(\mathcal {C})\).
Let G be a graph such that Open image in new window for all \(\hat{H} \in \hat{M}\). Suppose for contradiction that G is not a disjoint union of graphs from \(\mathcal {C}\). Then there is a component \(G'\) of G that is not in \(\mathcal {C}\) and therefore there is a graph \(H \in M(\mathcal {C})\) such that Open image in new window . We show that one of the graphs in the connection closure of H is a minor of \(G'\).
Let \(\{w_1, \dots , w_s\}\) be the vertex set of H and consider the image \(T_1, \dots , T_s\) of the minor map from H to \(G'\). Let T be a minimal subtree of \(G'\) that contains all of the \(T_i\). Such a tree must exist since \(G'\) is connected. Let \(\hat{H}\) be the graph with the same vertex set as H, and an edge between two vertices \(w_i, w_j\) whenever either \(w_iw_j \in E(H)\) or when there is a path between \(T_{w_i}\) and \(T_{w_j}\) in T that is disjoint from any \(T_{w_k}\) with \(w_i \ne w_k \ne w_j\). We claim that \(\hat{H}\) is in the connection closure of H. By construction, \(\hat{H}\) is connected and contains all components of H as disjoint subgraphs, so we only need to argue minimality. \(\hat{H}\) has no vertices besides those in H so no graph obtained by deleting a vertex would contain all components of H as subgraphs. To see that no edge of \(\hat{H}\) is superfluous, we note it has exactly \(e+m1\) edges and thus no proper subgraph could be connected and have all components of H as disjoint subgraphs. By the construction Open image in new window , so by the transitivity of the minor relation we have that Open image in new window .
Conversely let G be an arbitrary graph and assume that \(\hat{H} \in \hat{M}\) and Open image in new window . Because \(\hat{H}\) is connected, there is a connected component \(G'\) of G such that Open image in new window . Now there must be a graph \(H \in M(\mathcal {C})\) such that \(\hat{H}\) is in the connection closure of H, and since H is a subgraph of Open image in new window . Then, by the transitivity of the minor relation, Open image in new window and thus \(G' \not \in \mathcal {C}\). Therefore G is not a disjoint union of graphs from \(\mathcal {C}\).\(\square \)
Now our main theorem is established by a simple induction:
Theorem 6
There is a computable function which takes a set M of excluded minors characterising a minorclosed class \(\mathcal {C}\) and \(k \ge 0\) to the set \(M(\mathcal {C}_k)\).
Proof
The proof is by induction. For \(k = 0\), the set of minimal excluded minors of \(\mathcal {C}_0\) is \(M(\mathcal {C}_0) = M(\mathcal {C})\), which is given. For \(k > 0\), we have that \(\mathcal {C}_k = \overline{{\mathcal {C}_{k1}}^{\text {apex}}}\). By the induction hypothesis we can compute \(M(C_{k1})\), by Theorem 5 we can compute \(M({\mathcal {C}_{k1}}^{\text {apex}})\) and using Lemma 5 we can compute the connection closure of \(M({\mathcal {C}_{k1}}^{\text {apex}})\) to obtain \(M(\overline{{\mathcal {C}_{k1}}^{\text {apex}}}) = M(\mathcal {C}_k)\).\(\square \)
So by the Robertson–Seymour Theorem we have the following:
Corollary 1
Let \(\mathcal {C}\) be a minorclosed graph class. Then the problem Elimination Distance to Excluded Minors is \(\mathrm {FPT} \).
6 Conclusion
We are motivated by the study of the fixedparameter tractability of edit distances in graphs. Specifically, we are interested in edit distances such as the number of vertex or edge deletions, as well as more involved measures like elimination distance. Aiming at studying general techniques for establishing tractability, we establish an algorithmic metatheorem showing that any slicewise firstorder definable and slicewise nowhere dense problem is \(\mathrm {FPT} \). This yields, for instance, the tractability of counting the number of vertex and edge deletions to a class of bounded degree. As a second result, we establish that determining elimination distance to any minorclosed class is \(\mathrm {FPT} \), answering an open question of [3].
A natural open question raised by these two results is whether elimination distance to the class of graphs of degree d is \(\mathrm {FPT} \). When d is 0, this is just the treedepth of a graph, and this case is covered by our first result. For positive values of d, it is not clear whether elimination distance is firstorder definable. Indeed, a more general version of the question is whether for any nowhere dense and firstorder definable \(\mathcal {C}\), elimination distance to \(\mathcal {C}\) is \(\mathrm {FPT} \).
Another interesting case that seems closely related to our methods, but is not an immediate consequence is that of classes that are given by firstorder interpretations from nowhere dense classes of graphs. For instance, consider the problem of determining the deletion distance of a graph to a disjoint union of complete graphs. This problem, known as the cluster vertex deletion problem is known to be \(\mathrm {FPT} \) (see [21]). The class of graphs that are disjoint unions of cliques is firstorder definable but certainly not nowhere dense and so the method of Sect. 3 does not directly apply. However, this class is easily shown to be interpretable in the nowhere dense class of forests of height 1. Can this fact be used to adapt the methods of Sect. 3 to this class?
References
 1.Adler, I., Grohe, M., Kreutzer, S.: Computing excluded minors. In: SODA ’08: Proceedings of the Nineteenth Annual ACMSIAM Symposium on Discrete Algorithms. SIAM (2008)Google Scholar
 2.Bose, R.C.: Strongly regular graphs, partial geometries and partially balanced designs. Pac. J. Math. 13(2), 389–419 (1963)MathSciNetCrossRefMATHGoogle Scholar
 3.Bulian, J., Dawar, A.: Graph isomorphism parameterized by elimination distance to bounded degree. In: Parameterized and Exact Computation—9th International Symposium, IPEC: Wroclaw, Poland, September 10–12, 2014. Revised Selected Papers 2014, pp. 135–146 (2014)Google Scholar
 4.Cai, L.: Fixedparameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58, 171–176 (1996)MathSciNetCrossRefMATHGoogle Scholar
 5.Dawar, A.: Finite model theory on tame classes of structures, MFCS, Lecture Notes in Computer Science, vol. 4708, pp. 2–12. Springer (2007)Google Scholar
 6.Dawar, A.: Homomorphism preservation on quasiwide classes. J. Comput. Syst. Sci. 76(5), 324–332 (2010)MathSciNetCrossRefMATHGoogle Scholar
 7.Dawar, A., Grohe, M., Kreutzer, S.: Locally excluding a minor. In: Proceedings of 22nd IEEE Symposium on Logic in Computer Science, pp. 270–279 (2007)Google Scholar
 8.Diestel, R.: Graph Theory. Springer, Berlin (2000)MATHGoogle Scholar
 9.Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (2012)MATHGoogle Scholar
 10.Ebbinghaus, H.D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Berlin (1999)MATHGoogle Scholar
 11.Flum, J., Grohe, M.: Fixedparameter tractability, definability, and modelchecking. SIAM J. Comput. 31(1), 113–145 (2001)MathSciNetCrossRefMATHGoogle Scholar
 12.Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)MATHGoogle Scholar
 13.Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar Fdeletion: approximation, kernelization and optimal FPT algorithms. In: 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 470–479 (2012)Google Scholar
 14.Frick, M., Grohe, M.: Deciding firstorder properties of locally treedecomposable structures. J. ACM 48, 1184–1206 (2001)MathSciNetCrossRefMATHGoogle Scholar
 15.Gajarský, J., Hlinený, P., Obdrzálek, J., Ordyniak, S., Reidl, F., Rossmanith, P., Sanchez Villaamil, F., Sikdar, S.: Kernelization using structural parameters on sparse graph classes, Algorithms—ESA 2013—21st Annual European Symposium, pp. 529–540 (2013)Google Scholar
 16.Golovach, P.A.: Editing to a graph of given degrees. In: Cygan, M., Heggernes, P. (eds.) Parameterized and Exact Computation, pp. 196–207. Springer, Berlin (2014)Google Scholar
 17.Golovach, P.A.: Editing to a connected graph of given degrees. MFCS, vol. 8635, Chapter 28, pp. 324–335 (2014)Google Scholar
 18.Grohe, M., Kreutzer, S., Siebertz, S.: Deciding firstorder properties of nowhere dense graphs. In: STOC ’14: Proceedings of the 46th Annual ACM Symposium on Theory of Computing. ACM (2014)Google Scholar
 19.Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: distance from triviality. In: Downey, R., Fellows, M., Dehne, F. (eds.) Parameterized and Exact Computation, pp. 162–173. Springer, Berlin (2004)Google Scholar
 20.Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
 21.Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixedparameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47, 196–217 (2010)MathSciNetCrossRefMATHGoogle Scholar
 22.Marx, D.: Parameterized coloring problems on chordal graphs. Theory Comput. Sci. 351, 407–424 (2006)MathSciNetCrossRefMATHGoogle Scholar
 23.Mathieson, L.: The parameterized complexity of degree constrained editing problems. Ph.D. thesis, Durham University, Durham (2009)Google Scholar
 24.Mathieson, L.: Graph Editing Problems with Extended Regularity Constraints. CoRR abs/1406.4718cs.CC (2015)Google Scholar
 25.Mathieson, L., Szeider, S.: Editing graphs to satisfy degree constraints: a parameterized approach. J. Comput. Syst. Sci. 78(1), 179–191 (2012)MathSciNetCrossRefMATHGoogle Scholar
 26.Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. J. Discrete Algorithms 7(2), 181–190 (2009)MathSciNetCrossRefMATHGoogle Scholar
 27.Nesetril, J., Ossona de Mendez, P.: Sparsity–graphs, structures, and algorithms. Springer, Berlin (2012)Google Scholar
 28.Nesetřil, J., de Mendez, P.O.: Sparsity—Graphs, Structures, and Algorithms. Springer, Berlin (2012)MATHGoogle Scholar
 29.Niedermeier, R.: Invitation to FixedParameter Algorithms. Oxford University Press, Oxford (2006)CrossRefMATHGoogle Scholar
 30.Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. J. Comb. Theory Ser. B 92, 325–357 (2004)MathSciNetCrossRefMATHGoogle Scholar
 31.Stewart, I.A.: On the fixedparameter tractability of parameterized modelchecking problems. Inf. Process. Lett. 106(1), 33–36 (2008)MathSciNetCrossRefMATHGoogle Scholar
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