Subexponential algorithms for variants of the homomorphism problem in string graphs

Okrasa, Karolina; Rzążewski, Paweł

doi:10.1016/j.jcss.2019.12.004

Cited by 17 publications

(9 citation statements)

References 49 publications

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“…In this section we prove that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-hard on (P 2 + P 5 , P 6 )-free graphs. We do this by modifying the construction used in [29] for proving that these two problems are NP-complete on P 13 -free segment graphs.…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Both Feedback Vertex Set and Odd Cycle Transversal are polynomialtime solvable on permutation graphs [4], and thus on P 4 -free graphs. Recently, Okrasa and Rzążewski [29] proved that Odd Cycle Transversal is NP-complete on P 13free graphs. A small modification of their construction yields the same result for Connected Odd Cycle Transversal.…”

Section: Introductionmentioning

confidence: 99%

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On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

et al. 2020

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A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time on (sP1 + P3)-free graphs for every integer s ≥ 1. We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known for Feedback Vertex Set and Odd Cycle Transversal. We complement these results by proving that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-complete on (P2 +P5, P6)-free graphs.called Connected Vertex Cover, Connected Feedback Vertex Set and Connected Odd Cycle Transversal, respectively. Garey and Johnson [15] proved that Connected Vertex Cover is NP-complete even on planar graphs of maximum degree 4 (see, for example, [14,31,36] for NPcompleteness results for other graph classes). Grigoriev and Sitters [18] proved that Connected Feedback Vertex Set is NP-complete even on planar graphs with maximum degree 9. More recently, Chiarelli et al. [10] proved that Connected Odd Cycle Transversal is NP-complete even on graphs of arbitrarily large girth and on line graphs.As all three decision problems and their connected variants are NP-complete, we can consider how to restrict the input to some special graph class in order to achieve tractability. Note that this approach is in line with the aforementioned results in the literature, where NP-completeness was proven on special graph classes. It is also in line with with, for instance, polynomial-time results for Connected Vertex Cover by Escoffier, Gourvès and Monnot [12] (for chordal graphs) and Ueno, Kajitani and Gotoh [35] (for graphs of maximum degree at most 3 and trees).Just as in most of these papers, we consider hereditary graph classes, that is, graph classes closed under vertex deletion. Hereditary graph classes form a rich framework that captures many well-studied graph classes. It is not difficult to see that every hereditary graph class G can be characterized by a (possibly infinite) set F G of forbidden induced subgraphs. If |F G | = 1, say F = {H}, then G is said to be monogenic, and every graph G ∈ G is said to be H-free. Considering monogenic graph classes can be seen as a natural first step for increasing our knowledge of the complexity of an NP-complete problem in a systematic way. Hence, we consider the following research question:How does the structure of a graph H influence the computational complexity of a graph transversal problem for input graphs that are H-free?Note that different graph transversal problems may behave differently on some class of H-free graphs. However, the general strategy for obtaining complexity results is to first try to prove that the restriction to H-free graphs is NP-complete whenever H contains a cycle or the claw (the 4-vertex star). This is usually done by showing, respectively, that the probl...

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Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

et al. 2020

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“…Now, let us show that the algorithm from Lemma 3.2 can be used to solve LLSHom(C 4 ) in P t -free graphs. The proof of the following lemma is based on a similar argument used by Okrasa and Rzążewski [34] in the non-list case.…”

Section: Algorithm For F -Free Graphs For F ∈ Smentioning

confidence: 99%

“…The complexity of variants of the graph homomorphism problem in other hereditary graph classes has also been considered [7,15,34].…”

Section: Introductionmentioning

confidence: 99%

List Locally Surjective Homomorphisms in Hereditary Graph Classes

Dvořák¹,

Krawczyk²,

Masařík³

et al. 2022

Preprint

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A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V (G) to V (H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V (H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F -free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H, F ) the problem can be solved in subexponential time.We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F -free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of boundeddegree forests F , that might lead to some tractability results, is the family S consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P3, C4} are the only connected ones that allow for a subexponential-time algorithm in F -free graphs for every F ∈ S (unless the ETH fails).

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“…For instance, list variants can be expressed by giving negative revenues for forbidden assignments (see e.g. [21,28]). Also, our algorithms work in a slightly larger generality than stated above, see Section 5 for precise statements.…”

Section: Finding Large H-colorable Subgraphs In Hereditary Graph Classesmentioning

confidence: 99%

Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes

Chudnovsky¹,

King²,

Pilipczuk³

et al. 2021

SIAM J. Discrete Math.

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We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k = 2 is equivalent (by complementation) to Odd Cycle Transversal.We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved: in {P5, F }-free graphs in polynomial time, whenever F is a threshold graph; in {P5, bull}-free graphs in polynomial time; in P5-free graphs in time n O(ω(G)) ; in {P6, 1-subdivided claw}-free graphs in time n O(ω(G) 3 ) . Here, n is the number of vertices of the input graph G and ω(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P5-free and for {P6, 1-subdivided claw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs.Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P5-free graphs, if we allow loops on H.

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Subexponential algorithms for variants of the homomorphism problem in string graphs

Cited by 17 publications

References 49 publications

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

List Locally Surjective Homomorphisms in Hereditary Graph Classes

Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes

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