Alliances in graphs of bounded clique-width

Kiyomi, Masashi; Otachi, Yota

doi:10.1016/j.dam.2017.02.004

Cited by 15 publications

(11 citation statements)

References 16 publications

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“…A polynomial time algorithm for finding minimum defensive alliance in series parallel graph is presented in [13]. Fernau and Raible showed in [7] that the defensive, offensive and powerful alliance problems and their global variants are fixed parameter tractable when parameterized by solution size k. Kiyomi and Otachi showed in [17], the problems of finding smallest alliances of all kinds are fixed-parameter tractable when parameteried by the vertex cover number. The problems of finding smallest defensive and offensive alliances are also fixed-parameter tractable when parameteried by the neighbourhood diversity [11].…”

Section: Known Resultsmentioning

confidence: 99%

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Defensive Alliances in Graphs

Gaikwad¹,

Maity²

2021

Preprint

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A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbours are in S. We study the parameterzied complexity of the Defensive Alliance problem, where the aim is to find a minimum size defensive alliance. Our main results are the following: (1) The Defensive Alliance problem has been studied extensively during the last twenty years, but the question whether it is FPT when parameterized by feedback vertex set has still remained open. We prove that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; (2) the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP ⊆ NP/poly, (3) it does not admit 2 o(n) algorithm under ETH, and (4) the Defensive Alliance problem on circle graphs is NP-complete.

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Section: Known Resultsmentioning

confidence: 99%

“…The minimum size of a vertex cover in G is the vertex cover number of G, denoted by vc(G). Parameterized by vertex cover number vc, the Defensive Alliance problem is FPT [17] and in this section we prove the following kernelization hardness of the Defensive Alliance problem.…”

Section: No Polynomial Kernel Parameterized By Vertex Cover Numbermentioning

confidence: 95%

Defensive Alliances in Graphs

Gaikwad¹,

Maity²

2021

Preprint

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“…This means that no fixed-parameter tractable algorithm exists under the common complexity-theoretic assumption W[1] = FPT. Still, recent work has shown the problem to be solvable in polynomial time on graphs of bounded clique-width [24], which implies that there is a polynomial-time algorithm for graphs of bounded treewidth. Our result proves that, for any such algorithm, the degree of this polynomial must necessarily depend on the treewidth unless W[1] = FPT.…”

Section: Resultsmentioning

confidence: 99%

“…The main contribution of this paper is a parameterized complexity analysis of Defensive Alliance with treewidth as the parameter. The question of whether or not this problem is fixed-parameter tractable when parameterized by treewidth has so far been unresolved [24]. In the current chapter, we provide a negative answer to this question: We show that the problem is hard for the class W [1], which rules out fixed-parameter tractable algorithms under commonly held complexity-theoretic assumptions.…”

Section: Introductionmentioning

confidence: 82%

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Defensive alliances in graphs of bounded treewidth

Bliem

Woltran

2018

Discrete Applied Mathematics

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A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbors is in S. The problem of finding a defensive alliance of minimum size in a given graph is NP-hard and there are polynomial-time algorithms if certain parameters are bounded by a fixed constant. In particular, fixed-parameter tractability results have been obtained for some structural parameters such as the vertex cover number. However, for the parameter treewidth, the question of whether the problem is FPT has remained open. This is unfortunate because treewidth is perhaps the most prominent graph parameter and has proven successful for many problems. In this work, we give a negative answer by showing that the problem is W[1]-hard when parameterized by treewidth, which rules out FPT algorithms under common assumptions. This is surprising since the problem is known to be FPT when parameterized by solution size and "subset problems" that satisfy this property usually tend to be FPT for bounded treewidth as well. We prove W[1]-hardness by using techniques from a recent hardness result for the problem of finding so-called secure sets in a graph.

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Alliances and Related Domination Parameters

Haynes

Hedetniemi

2012

Developments in Mathematics

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Alliances in graphs of bounded clique-width

Cited by 15 publications

References 16 publications

Defensive Alliances in Graphs

Defensive Alliances in Graphs

Defensive alliances in graphs of bounded treewidth

Alliances and Related Domination Parameters

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