Distance paired domination numbers of graphs

Raczek, Joanna

doi:10.1016/j.disc.2007.05.018

Cited by 12 publications

(5 citation statements)

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“…Proof. We prove this by providing a reduction from the W[2]-complete problem Dominating Set that is inspired by [22,Theorem 7]. The input to this problem is a graph X = (V, E) and a number k (treated as parameter) and the question is whether there exists a dominating set D ⊆ V of size k in X, meaning that each vertex v ∈ V \ D is adjacent to at least one vertex in D. We transform the Dominating Set instance (X, k) with X = (V, E) into an equivalent instance (X , k) where X = (V , E , c ) for k-Discrete[ ].…”

Section: :10mentioning

confidence: 99%

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Arvind

Fuhlbrück²,

Köbler³

et al. 2022

ACM Trans. Comput. Theory

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In this paper we study the algorithmic complexity of the following problems: (1) Given a vertex-colored graph X = ( V , E , c ), compute a minimum cardinality set of vertices S ⊆ V such that no nontrivial automorphism of X fixes all vertices in S . A closely related problem is computing a minimum base S for a permutation group \(G\le \operatorname{Sym}(n) \) given by generators, i.e., a minimum cardinality subset S ⊆[ n ] such that no nontrivial permutation in G fixes all elements of S . Our focus is mainly on the parameterized complexity of these problems. We show that when \(k=|S|\) is treated as parameter, then both problems are \(\textsf {MINI[1]} \) -hard. For the dual problems, where \(k=n-|S|\) is the parameter, we give FPT algorithms. (2) A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure.

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Section: :10mentioning

confidence: 99%

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Arvind

Fuhlbrück²,

Köbler³

et al. 2022

ACM Trans. Comput. Theory

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“…The cardinality of a smallest S γ k p (G), is known as the k-distance paired domination number, γ k p (G), of a graph (See [14] for details). Theorem 5.5: If G is a connected graph, then,…”

Section: B Eternal 1-security and The K-distance Paired Dominationmentioning

confidence: 99%

Securing multiagent systems against a sequence of intruder attacks

Abbas

Egerstedt

2012

2012 American Control Conference (ACC)

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In this paper, we discuss the issue of security in multiagent systems in the context of their underlying graph structure that models the interconnections among agents. In particular, we investigate the minimum number of guards required to counter an infinite sequence of intruder attacks with a given sensing and response range of an individual guard. We relate this problem of eternal security in graphs to the domination theory in graphs, providing tight bounds on the number of guards required along with schemes for securing a multiagent system over a graph.

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“…The preceding proof is inspired by [22,Theorem 7] describing an fixed parameter reduction from Dominating Set to a problem called d-Distance Paired Dominating Set, which asks for a given graph and a number k (treated as parameter) whether there is a set C of k vertices such that all vertices in the graph are within distance d of a vertex in C and there is a perfect matching between the vertices in C.…”

Section: Bounded Number Of Refinement Stepsmentioning

confidence: 99%

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Arvind¹,

Fuhlbrück²,

Köbler³

et al. 2016

Preprint

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In this paper we study the complexity of the following problems:1. Given a colored graph X = (V, E, c), compute a minimum cardinality set of vertices S ⊂ V such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G ≤ Sn given by generators, i.e., a minimum cardinality subset S ⊂ [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k = |S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k = n − |S| is the parameter, we give FPT algorithms. 2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c) compact, or (d) refinable.In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time (even in logspace), while starting from color class size 4 they become W[P]-hard.

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Distance paired domination numbers of graphs

Cited by 12 publications

References 1 publication

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Securing multiagent systems against a sequence of intruder attacks

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

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