Linear Kernels and Linear-Time Algorithms for Finding Large Cuts

The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of those results:We show that an algorithm by Crowston et al. (Algorithmica 72(3):734–757, 2015) for (Signed) Max-Cut Above Edwards−Erd ő s Bound can be implemented so as to run in linear time $$8^k\cdot O(m)$$ 8k·O(m) ; this significantly improves the previous analysis with run time $$8^k\cdot O(n^4)$$ 8k·O(n4) .We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards−Erd ő s Bound with O(k) vertices, improving a kernel with $$O(k^3)$$ O(k3) vertices by Crowston et al. (Theor Comput Sci 513:53–64, 2013).We improve all known kernels for parameterizations above strongly $$\lambda $$ λ -extendible properties (a generalization of the Max-Cut results) by Crowston et al. (Proceedings of FSTTCS 2013, Leibniz international proceedings in informatics, Guwahati, 2013) from $$O(k^3)$$ O(k3) vertices to O(k) vertices.Therefore, Max Acyclic Subdigraph parameterized above Poljak–Turzík bound admits a kernel with O(k) vertices and can be solved in $$2^{O(k)}\cdot n^{O(1)}$$ 2O(k)·nO(1) time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad, 2012). All presented kernels can be computed in time O(km). We show that an algorithm by Crowston et al. (Algorithmica 72(3):734–757, 2015) for (Signed) Max-Cut Above Edwards−Erd ő s Bound can be implemented so as to run in linear time $$8^k\cdot O(m)$$ 8k·O(m) ; this significantly improves the previous analysis with run time $$8^k\cdot O(n^4)$$ 8k·O(n4) . We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards−Erd ő s Bound with O(k) vertices, improving a kernel with $$O(k^3)$$ O(k3) vertices by Crowston et al. (Theor Comput Sci 513:53–64, 2013). We improve all known kernels for parameterizations above strongly $$\lambda $$ λ -extendible properties (a generalization of the Max-Cut results) by Crowston et al. (Proceedings of FSTTCS 2013, Leibniz international proceedings in informatics, Guwahati, 2013) from $$O(k^3)$$ O(k3) vertices to O(k) vertices. Therefore, Max Acyclic Subdigraph parameterized above Poljak–Turzík bound admits a kernel with O(k) vertices and can be solved in $$2^{O(k)}\cdot n^{O(1)}$$ 2O(k)·nO(1) time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad, 2012).


Introduction
A recent paradigm in parameterized complexity is to not only show a problem to be fixedparameter tractable, but indeed to give algorithms with optimal run times in both the parameter and the input size.Ideally, we strive for algorithms that are linear in the input size, and optimal in the dependence on the parameter k assuming a standard hypothesis such as the Exponential Time Hypothesis [17].New results in this direction include fixed-parameter algorithms for Graph Bipartization [18,30], Planar Subgraph Isomorphism [9], DAG Partitioning [29] and Subset Feedback Vertex Set [20].
Here, we consider the fundamental Max-Cut problem from the view-point of linear-time fixed-parameter algorithms.In this classical NP-complete problem [19], the task is to find a Poljak and Turzík [25] showed that the property of having a large cut (i.e., a large bipartite subgraph) can be generalized to many other classical graph properties, including properties of oriented and edge-labeled graphs.They defined the notion of "λ-extendable" properties Π and generalized the lower bound (1) to tight lower bounds for all such properties; we refer to these lower bounds as the Poljak-Turzík bound for Π.Well-known examples of such properties include bipartite subgraphs, q-colorable subgraphs for fixed q, or acyclic subgraphs of oriented graphs.Mnich et al. [23] considered the problem Above Poljak-Turzík(Π) of finding subgraphs in Π with k edges above the Poljak-Turzík bound; they gave fixed-parameter algorithms for this problem on all "strongly" λ-extendable properties Π.A subclass of these, requiring certain technical conditions, was later shown to admit polynomial kernels [8].

Our Contributions
Linear-Time FPT.Our first result is that the fixed-parameter algorithm given by Crowston et al. [4] for the Signed Max-Cut AEE problem can be implemented in such a way that it runs in linear time.
Theorem 1 considerably improves the earlier run time analysis [4,7], which shows a run time of 8 k • O(n 4 ).At the same time, our algorithm improves the very involved algorithm by Bollobás and Scott [2] that considers the weaker lower bound m/2 + ( √ 8m + 1 − 1)/8 instead of (1).Third, Theorem 1 generalizes the linear-time algorithm by Ngo .c and Tuza [24] for the special case of Max-Cut with k = 0. Note that Max-Cut AEE cannot be solved in time 2 o(k) • n O (1) assuming the Exponential Time Hypothesis [7].
Linear Vertex Kernels.Our second contribution is a kernel with a linear number O(k) of vertices for Max-Cut AEE and its generalization Signed Max-Cut AEE.These results considerably improve the previous best kernel bound of O(k 3 ) vertices by Crowston et al. [4].Moreover, the presented kernel completely resolves the asymptotic kernelization complexity of (Signed) Max-Cut AEE, since a kernel with o(k) vertices would again contradict the Exponential-Time Hypothesis, as the Max-Cut problem can be solved by checking all vertex bipartitions.On top of that, our kernelization is also fast.In fact, we only need to compute O(k) DFS/BFS trees.The rest of the algorithm runs in time O(m + n).

Extensions to Strongly λ-Extendable
Properties.As mentioned, the property of graphs having large bipartite subgraphs can be generalized to λ-extendable properties as defined by Poljak and Turzík [25] (we defer the formal definitions to Section 2).For a given λ-extendable property Π, we consider the following problem.

Input:
A connected graph G and an integer k.
Note the slight change in the definition of k compared to (Signed) Max-Cut AEE, where k was divided by 4 = 2 1−λ for λ = 1 2 .Crowston et al. [4] gave polynomial kernels with O(k 3 ) or O(k 2 ) vertices for the problem Above Poljak-Turzík(Π), for all strongly λ-extendable properties Π on possibly oriented and/or labeled graphs satisfying at least one of the following properties.(P1) λ = 1 2 ; or (P2) G ∈ Π for all graphs G whose underlying simple graph is K 3 ; or (P3) Π is a hereditary property of simple or oriented graphs.Our third result improves all these kernels for strongly λ-extendable properties to asymptotically optimal O(k) vertices:

Theorem 3. Let Π be any strongly λ-extendable property of (possibly oriented and/or labeled) graphs satisfying (P1), or (P2), or (P3). Then Above Poljak-Turzík(Π) admits a kernel with O(k) vertices, which is computable in time O(km).
Consequences for Acyclic Subdigraphs.Theorem 3 has several applications.For instance, Raman and Saurabh [27] asked for the parameterized complexity of the Max Acyclic Subdigraph problem above the Poljak-Turzík bound: Given a weakly connected oriented graph G on n vertices and m arcs, does it have an acyclic sub-digraph of at least m/2 + (n − 1)/4 + k arcs?For this problem, Crowston et al. [6] gave an algorithm with run time 2 O(k log k) • n O (1) and showed a kernel with O(k 2 ) vertices.They explicitly asked whether the kernel size can be improved to O(k) vertices, and whether the run time can be improved to 2 O(k) • n O (1) .Here, we answer their questions in the affirmative by using Theorem 3 and  O(k) • n O (1) .
Again, assuming the Exponential Time Hypothesis, the run time of this algorithm is asymptotically optimal.
Due to space constraints, proofs of statements marked by ( ) are deferred to the full version.

Preliminaries
We use to denote the disjoint union of sets.A clique tree is a connected graph whose blocks are cliques, where a clique is a complete subgraph of a graph.A clique forest is a graph whose connected components are clique trees. 1 For an oriented and/or labeled graph G we say that G has one of the above-defined properties if G does.
Let G be a graph.For denote the set of edges with one endpoint in V 1 and the other endpoint in V 2 .For signed graphs G, let E + (G) ⊆ E(G) be the edges with positive labels, and A graph property Π is simply a set of graphs.For a graph G, a Π-subgraph is a subgraph of G that belongs to Π.A graph property Π is hereditary if for any G ∈ Π also all vertex-induced subgraphs of G belong to Π. Poljak and Turzík [25] defined the notion of "λ-extendability" for graph properties Π, and proved a lower bound on the size of any Π-subgraph in arbitrary graphs.A related notion of "strong λ-extendability" was introduced by Mnich et al. [23]; any strongly λ-extendable property is λ-extendable, but it is unclear whether the other direction holds.
Definition 5. Let G be a class of (possibly labeled and/or oriented) graphs and let λ ∈ (0, 1).A (graph) property Π is strongly λ-extendable on G if it satisfies the following properties: (ii) block additivity: G ∈ G belongs to Π if and only if each block of G belongs to Π.
(iii) extendability: For any G ∈ G and any partition The set of all bipartite graphs Π bipartite is a strongly 1 2 -extendable property.Thus, Max-Cut AEE is equivalent to Above Poljak-Turzík Bound(Π bipartite ).
Poljak and Turzík [25] showed that, given a (strongly) λ-extendable property Π, any connected graph G contains a subgraph edges such that H ∈ Π.We denote this lower bound by pt(G).Further, we define the excess of G over this lower bound with respect to Π as When considering properties of labeled and/or oriented graphs, we denote by ex(K t ) the minimum value of ex(G) over all labeled and/or oriented graphs G with G = K t ; here, K t denotes the complete graph of order t.(Our definition slightly differs from the one by Crowston et al. [8].) A strongly λ-extendable property Π diverges on cliques if ex(K j ) > 1−λ 2 for some j ∈ N.For example, every strongly λ-extendable property with λ = 1 2 diverges on cliques [8].We recall the following fact about diverging properties: Lemma 8]).Let Π be a strongly λ-extendable property diverging on cliques, and let j ∈ N, a > 0 be such that ex(K j ) = 1−λ 2 + a. Then ex(K i ) ≥ ra for all i ≥ rj.We need the following proposition in all sections.For Signed Max Cut, we will apply it with λ = 1 2 .Proposition 7 ([8, Lemma 6]).Let Π be a strongly λ-extendable property, let G be a connected graph and let

Linear-Time Fixed-Parameter Algorithms and Linear Vertex Kernels for Signed Max Cut
In this section we consider the Signed Max-Cut AEE problem.We show that the fixedparameter algorithm given by Crowston et al. [4]

Linear-Time Fixed-Parameter Algorithm
The algorithm by Crowston et al. [4] starts by applying the following seven reduction rules.We restate them here, as they are crucial for our results.A reduction rule is 1-safe if, on input I S A A C 2 0 1 6

31:6
Linear Kernels and Linear-Time Algorithms for Finding Large Cuts (G, k) it returns a pair (G , k ) such that (G, k) is a "yes"-instance for Signed Max-Cut AEE if (G , k ) is. (Note that the converse direction does not have to hold.)In a signed graph G we call a triangle positive if its number of negative edges is even.In the description of the rules, G is always a connected signed graph and C is always a clique that does not contain a positive triangle.

Reduction Rule 13. If there is a vertex
We subsume the results by Crowston et al. [4] in the following proposition.Proposition 17 ([4]).Rules 9-15 are 1-safe.To any connected signed graph with at least one edge, one of these rules applies and the resulting graph is connected.If S is the set of vertices marked during the exhaustive application of Rules 9-15 on a connected signed graph G, then G − S is a clique forest.If |S| > 3k, then (G, k) is a "yes"-instance.Given an instance (G, k), we can thus compute in time O(k • |E(G)|) a vertex set S that either proves that (G, k) is a "yes"-instance or G − S is a clique forest.We now show that, if a partition for the vertices in S is already given, we can in time O(|E(G)|) compute an optimal extension to G. We use the following problem, which goes back to Crowston et al. [7].

Max-Cut Extension
Input: A clique forest GS with weight functions wi : V (GS) → N0 for i = 0, 1. Task: Find an assignment ϕ : We now give a proof sketch for Theorem 1. Lemma 18 allows us to find the set S from Proposition 17 in time O(km) (the case that k is not decreased can only take O(m) total time).Guess one of the at most 2 3k partitions on S and solve the corresponding Max-Cut Extension problem with Lemma 19.

A Linear Vertex Kernal for Signed Max-Cut AEE
For the whole section, let G 0 be the original graph, let S be the set of marked vertices during the exhaustive application of Rules 9-15 on G 0 , and let G r be the resulting graph after the exhaustive application of our kernelization Rules 20-21 (to be defined later) on G 0 .
If there is a (unique by Proposition 17) remaining vertex v left after the exhaustive application of Rules 9-15, then add a path vwx to G, i.e., define G = (V (G) ∪ {w, x}, E(G) ∪ {vw, wx}).Then (G , k + 2) is an instance of Max-Cut AEE that is due to Proposition 16 equivalent to (G, k) because the excess of a path of length 2 is 2/4.This implies that we can w.l.o.g.assume that every vertex gets removed during the exhaustive application of the reduction rules because we can assume we finish with deleting the new path with Rule 14.Furthermore, as Rule 13 can then not be applied last, we can assume that at least one of the vertices that are removed last is contained in S.
We will use two-way reduction rules which are similar to the two-way reduction rules by Crowston et al. [4].However, our two-way reduction rules have the property that connected components of G − S cannot fall apart, i.e., two blocks in G r − S are reachable from each other if and only if the corresponding blocks in G 0 − S are reachable from each other.We can thus show that Rules 9-15 can behave "equivalently" on G r as on G 0 (Lemma 24), i.e., that the same set S of vertices can also be marked in G r .This is the crucial idea which allows us to obtain better kernelization results than previous papers, as it allows the following analysis.
To show size bounds for our kernel G r , we (hypothetically) change the set of rules in such a way that whenever a vertex s ∈ S is about to be removed, we additionally remove internal vertices from different blocks of G r − S that are all adjacent to s.This means that for every s ∈ S, we find a star-like structure Y s such that Y s is removed together with s, and the excess on Y s grows linearly in |Y s |.We can distribute the internal vertices from G − S in such a way to the different Y s that all generated graphs are still connected.Then the large excess of the different Y s translates to a large excess of G r through Proposition 7.
We use this approach twice to first bound the number of special blocks (Lemma 25) and then the number of internal vertices in special blocks (Lemma 27) to O(k).On the other I S A A C 2 0 1 6

31:8
Linear Kernels and Linear-Time Algorithms for Finding Large Cuts hand, due to Rules 20-21 a constant fraction of vertices in G r − S must be adjacent to S. This completes the proof.
Let C be a block in the clique forest G − S.
is non-empty.Let B be the set of blocks and B s be the set of special blocks in G r − S. A block C is a ∆-block if it is not special, contains exactly three vertices, and We now give our two-way reduction rules, which on input (G, k) produce an instance (G , k) of Signed Max-Cut AEE.Note that the parameter k does not change.We call a rule 2-safe if (G, k) is a "yes"-instance if and only if (G , k) is.The first rule is again due to Crowston et al. [4], who showed it to be 2-safe.The run time analysis is our work.Recall our assumption that (without loss of generality) G − S does not contain positive edges.

Reduction Rule 20. Let C be a block in
Lemma 22 ( ).Rules 20-21 are 2-safe.If they are applied to a connected graph G, then the resulting graph G is also connected.

Lemma 23 ( ). Given S, Rules 20-21 can be applied exhaustively to G 0 in total time O(m+ n).
Lemma 24 ( ).Rules 9-15 can be applied exhaustively to the graph G r in such a way that the set S of marked vertices is equal to S.Moreover, if only the Rules 11/13/14/15 are applied to G 0 , the same set of rules is applied to G r .
The last part of the lemma will be needed later in Section 4.2.

Lemma 25 ( ).
If G r −S has more than 11k special blocks, then (G r , k) is a "yes"-instance of Signed Max-Cut AEE.

Lemma 26 ( ).
If G r − S has more than 48k blocks, then (G r , k) is a "yes"-instance of Signed Max-Cut AEE.Otherwise, G r − S has at most 48k external vertices, and Lemma 27 ( ).If there are more than 117k internal vertices in special blocks in G r − S, then (G r , k) is a "yes"-instance of Signed Max-Cut AEE.
We are now ready to prove Theorem 2. ) to obtain an equivalent instance (G r , k).Check whether (G r , k) is a "yes"-instance due to Lemma 26 or Lemma 27.If this is not the case, then there are at most 3k vertices in S, at most 48k external vertices in G r − S and at most 117k internal vertices in special blocks.If there were more internal than external vertices in a non-special block, we could apply Rule 20 to this block.Thus, the number of internal vertices in non-special blocks is bounded by 96k according to Lemma 26.Hence, the total number of vertices in G r is bounded by 3k + 48k + 117k + 96k = 264k.

4
Linear Vertex Kernels for λ-Extendable Properties In this section we extend our linear kernels for Signed Max-Cut to all strongly λ-extendable properties satisfying (P1), or (P2), or (P3).Henceforth, fix a strongly λ-extendable property Π, and let (G 0 , k) be an instance of Above Poljak-Turzík Bound(Π).For notational brevity, we assume the empty graph to be in Π.
As in the previous section, we use a set of 1-safe reduction rules devised by Mnich et al. [23] to find a set S such that G 0 − S is a clique forest; the difference compared to Signed Max-Cut is the different change of k.Since we change the reduction rules slightly in the next section, we refrain from stating the rules by Mnich et al. here.

Lemma 28 ([23]
).There is an algorithm that, given a connected graph G and k ∈ N, either decides that ex(G) ≥ k, or finds a set S of at most 6k 1−λ vertices such that G − S is a clique forest.This also holds for all strongly λ-extendable properties of oriented and/or labeled graphs.
The detection which of the reduction rules can be applied to a graph G is completely analogous to the Signed Max-Cut reduction rules.Hence, it follows immediately from Lemma 18 that the rules can be applied exhaustively in time O(km).

Linear Kernel for Properties Diverging on Cliques
We show that Above Poljak-Turzík Bound(Π) admits kernels with O(k) vertices for all strongly λ-extendable properties Π that are diverging on cliques and for which ex(K i ) > 0 for all i ≥ 2.

Strongly 1 2 -Extendable Properties on Oriented Graphs
We now turn to strongly 1 2 -extendable properties Π on oriented graphs.First of all we modify the reduction rules by Mnich et al. [23] in such a way that they are compliant with Rules 9-15.Let G always be a connected graph.Linear Kernels and Linear-Time Algorithms for Finding Large Cuts Rules 31-34 are exactly Rules 13/11/14/15 for Signed Max-Cut AEE with all edges negative.

Reduction Rule 31. Let
Lemma 35 ( ).Rules 31-34 are 1-safe.To any connected graph with at least one edge, one of the rules applies and the resulting graph is connected.If S is the set of marked vertices, then G − S is a clique forest.If |S| > 12k, then (G, k) is a "yes"-instance.
Like Crowston et al. [8], we restrict ourselves to hereditary properties.Let → K3 be the orientation of K 3 which is an oriented cycle, and let K3 be the only (up to isomorphisms) other orientation of K 3 .Crowston et al. [8] showed that if → K3∈ Π, then also K3∈ Π, and thus Theorem 30 applies.We now consider the case that → K3 ∈ Π together with K3∈ Π. Proposition 36 ([8]).Let Π be a hereditary strongly 1  2 -extendable property on oriented graphs with K3∈ Π.Then ex(K i ) > 0 for all i ≥ 4 and Π diverges on cliques.
Following this lemma, the conditions of Lemma 29 are almost satisfied.The only oriented cliques without positive excess are K 1 and → K3, because ex(K 2 ) = 1  4 for 1 2 -extendable properties.Blocks isomorphic to K 1 can only occur as isolated vertices in G − S. We can bound these like in the previous section.Hence, we only need reduction rules to bound the number of blocks B in a clique forest with B ∼ = → K3.Let Π be a hereditary strongly 1  2 -extendable property on oriented graphs with K3∈ Π.Let (G 0 , k) be an instance of Above Poljak-Turzík(Π).Lemma 35 either proves that (G 0 , k) is a "yes"-instance, or it finds a set S of at most 12k vertices such that G 0 − S is a clique forest.Starting with (G 0 , k), we apply the following reduction rules, which on input (G, k) produce an equivalent instance (G , k).

Reduction Rule 37. Delete
, where w i is the internal vertex of B i .Delete v 3 and w 3 .Add edges v 2 v 4 and w 2 v 4 .
Intuitively speaking, Rule 38 takes three blocks in G − S that form a "path" and are all isomorphic to → K3.If all vertices except the "endpoints" v 1 and v 4 are not adjacent to S, then it is safe to delete one block.For an illustration, see Fig. 1.
From now on, let G r be the resulting graph after the exhaustive application of Rules 37-38 on G 0 .Rules 37-38 are special cases of Rules 20-21.Because Rules 31-34 are Rules 13/11/14/15 for Signed Max-Cut AEE with all edges negative, the next lemma follows from Lemma 24.
Lemma 40.Rules 31-34 can be applied exhaustively on the graph G r in such a way that the set S of vertices removed by their application is equal to S.
Let B + be the set of blocks of G r − S with positive excess, and let B − be the other blocks, i.e., the blocks Further, let V + ⊆ V (G) \ S be the set of vertices in blocks with positive excess, V − be the set of vertices in blocks from B − , and let V − int V − ext = V − be the set of internal and external vertices of blocks B ∈ B − , respectively.Note that V + and V − may intersect.

Lemma 41 ( ). It holds |V
Using the same approach as in Section 4.1, one can show that |V + | = O(k) or (G r , k) is a "yes"-instance.As Lemma 41 bounds |V − | = O(k) for every "no"-instance, and V + ∪ V − ∪ S = V (G r ), this suffices to prove the following result.
Proof of Theorem 3. Let λ ∈ (0, 1) and let Π be a strongly λ-extendable property of (possibly oriented and/or labeled) graphs.If λ = 1 2 or G ∈ Π for every G with G = K 3 , we can use Theorem 30.Otherwise, we only have to consider the case that Π is a hereditary property of simple or oriented graphs.
Consider the case that → K3∈ Π or K3∈ Π.If → K3∈ Π, then Crowston et al. [8] show that K3∈ Π, i.e., we can use Theorem 30.And if K3∈ Π, we use Theorem 42.Now we may suppose that G ∈ Π for every G with G = K 3 .Then Crowston et al. [8] show that Π is the set of all bipartite graphs.Hence, in the case of simple graphs as well as if → K3, K3 ∈ Π for oriented graphs, we can use Theorem 2 to obtain a linear vertex kernel.
It is easy to see that Rules 37-38 can be applied exhaustively in time O(m).As λ is constant and we can apply every other reduction rule in linear time, it follows a total run time of O(λ • km) = O(km).

Discussion
For the classical (Signed) Max-Cut problem, and its wide generalization to strongly λ-extendable properties, parameterized above the classical Poljak-Turzík bound, we improved the run time analysis for a known fixed-parameter algorithm to 8 k • O(m).We further improved all known kernels with O(k 3 ) vertices for these problems to asymptotically optimal O(k) vertices.We did not try to optimize the hidden constants, as the analysis is already quite cumbersome.
It remains an interesting question whether all positive results presented here extend to edge-weighted graphs, where each edge receives a positive integer weight and the number m of edges in the Edwards-Erdős bound (1) is replaced by the total sum of the edge weights.I S A A C 2 0 1 6

31:12
Linear Kernels and Linear-Time Algorithms for Finding Large Cuts Further, Mnich et al. [23] showed fixed-parameter tractability of Above Poljak-Turzík Bound(Π) for all strongly λ-extendable properties Π.However, the polynomial kernelization results by Crowston et al. [8] as well as in this paper do not seem to apply to the special case of non-hereditary 1  2 -extendable properties.Such properties Π exist; e.g., Π = {G ∈ G | C ∼ = K 3 for all 2-connected components C of G}.Also, for 1 2 -extendable properties on labeled graphs we only showed a polynomial kernel for the special case of Signed Max-Cut.It would be desirable to avoid these restrictions.

Theorem 2 .
The (Signed) Max-Cut AEE problem admits a kernel with O(k) vertices, which can be computed in time O(km).

Rule 9 .
If abca is a positive triangle such that G − {a, b, c} is connected, then mark a, b, c, delete them, and set k = k − 3. Reduction Rule 10.If abca is a positive triangle such that G − {a, b, c} has exactly two connected components C and Y , then mark a, b, c, delete them, delete C, and set k = k − 2. Reduction Rule 11.Let C be a connected component of G − v for some vertex v ∈ V (G).If there exist a, b ∈ V (C) such that G − {a, b} is connected and there is an edge av but no edge bv, then mark a, b, delete them, and set k = k − 2. Reduction Rule 12. Let C be a connected component of G − v for some vertex v ∈ V (G).If there exist a, b ∈ C such that G − {a, b} is connected and vabv is a positive triangle, then mark a, b, delete them, and set k = k − 4.
then mark a, b, c, delete them, and set k = k − 1. Reduction Rule 15.Let C, Y be the connected components of G − {v, b} for some vertices v, b ∈ V (G) such that vb / ∈ E(G).If G[V (C) ∪ {v}] and G[V (C) ∪ {b}] are cliques that do not contain a positive triangle, then mark v, b, delete them, delete C, and set k = k − 1.We slightly changed Rule 13.Crowston et al. [4] always set k = k, whereas we set k = k−1 when |V (C)| is odd.In this case, pt(G[V (C) ∪ {v}]) cannot be integral because |V (C) ∪ {v}| is even, and thus ex(G[V (C) ∪ {v}]) ≥ 1 4 .Therefore our change for k is 1-safe due to the following result.Proposition 16 ([4,Lemma 2]).Let G be a connected signed graph and Z be a connected

Following
Crowston et al. [4, Corollary 3], we assume -without loss of generality -from now on that the resulting clique forest G − S does not contain a positive edge.Lemma 18 ( ).Let G be a connected signed graph, let X be a leaf block of G, and let r ∈ V (G) such that V (X) \ {r} does not contain a cut vertex of G. Then we can apply one of the Rules 9-15 to G deleting and marking only vertices from X in time O(|E(X)|).

Theorem 2 .
Let (G 0 , k) be an instance of Signed Max-Cut AEE.Like in Section 3.1, apply Rules 9-15 exhaustively to (G 0 , k) in time O(k •|E(G 0 |), producing an instance (G , k ) and a vertex set S of marked vertices.If k ≤ 0, then (G , k ) and thus also (G, k) is a "yes"-instance.Now apply Rules 20-21 exhaustively to (G 0 , k) in time O(|E(G)|) (Lemma 23

2 .
is a clique.Delete C and set k = k.Reduction Rule 32.Let C be a connected component of G − v for some vertex v ∈ V (G) such that C is a clique.If there exist a, b ∈ V (C) such that G − {a, b} is connected and av ∈ E(G), but bv / ∈ E(G), then mark a, b, delete them, and set k = k − 1

2 0 1 6 31:4 Linear Kernels and Linear-Time Algorithms for Finding Large Cuts then
applying an O * (2 n )-time algorithm by Raman and Saurabh [28, Thm.2] to our kernel with O(k) vertices.
We sometimes write an edge e = {u, v} as e = uv, if no confusion arises; this way, three distinct vertices a, b, c can induce a triangle abca.In a labeled graph, each edge in E(G) receives one of a constant number of labels.For an oriented and/or labeled graph G, let G denote the underlying simple graph obtained from omitting orientations and/or labels.Throughout the paper, we assume graphs to be encoded as adjacency lists.A graph is connected if there is a path between any two of its vertices.A connected component of G is a maximal connected subgraph of G.A cut vertex of a graph G is a vertex whose removal increases the number of connected components.A graph is 2-connected if it does not contain any cut vertices.A maximal 2-connected subgraph of a graph G is called a block of G.A block that contains at most one cut vertex of G is called a leaf block of G.