Paired-domination of Cartesian products of graphs and rainbow domination

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. ᭧

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“…Brešar et al [8] established the following result on the total domination number of the Cartesian product G H.…”

Section: T (G) T (H) If and Only If T (G) = 2 (G) And H Consists Ofmentioning

confidence: 97%

“…A version of Vizing's Conjecture for the paired-domination number is studied in [8]. For this purpose, recall that for k ≥ 2, a k-packing in a graph G was defined by Meir and Moon [37] as a set S of vertices of G that are pairwise at distance greater than…”

Section: Paired-dominationmentioning

confidence: 99%

Vizing's conjecture: a survey and recent results

Brešar¹,

Dorbec²,

Goddard³

et al. 2011

Journal of Graph Theory

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124

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“…For example, in Figure 3, the edges e 1 and e 2 are redundant in light of e 3 . We note that we are interested in minimizing the non-covered part of the plane rather than minimizing the number of points that are not dominated.…”

Section: Definition 31 Let G Be An Embedded Permutation Graph Q An mentioning

confidence: 99%

“…, π i k ) is called a left-to-right maxima subsequence if it consists of all the left-to-right maxima of π; clearly, π i1 = π 1 . For example, the left-to-right maxima subsequence of the permutation (4,2,6,1,9,3,7,5,12,11,8,10) is (4,6,9,12).…”

Section: Theoretical Frameworkmentioning

confidence: 99%

“…Kang et al [16] considered "inflated" graphs (for a graph G, its inflated version is obtained from G by replacing each vertex of degree d in G by a clique on d vertices), gave an upper and a lower bound for the paired-domination number of the inflated version of a graph, and described a linear-time algorithm for computing a minimum paired-dominating set of an inflated tree. Bounds for the paired-domination number have been established also for claw-free cubic graphs [8], for cartesian products of graphs [3], and for generalized claw-free graphs [6]. Very recently, Cheng et al [5] gave an O(n + m)-time algorithm for computing a paired-dominating set of an interval graph on n vertices and m edges, when an interval model for the graph with endpoints sorted is available; they also extended their result to circular-arc graphs giving an algorithm running in O(m(m + n)) time in this case.…”

Section: Introductionmentioning

confidence: 99%

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An O(n)-Time Algorithm for the Paired-Domination Problem on Permutation Graphs

Lappas

Nikolopoulos

Palios

2009

Lecture Notes in Computer Science

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A vertex subset D of a graph G is a dominating set if every vertex of G is either in D or is adjacent to a vertex in D. The paired-domination problem on G asks for a minimum-cardinality dominating set S of G such that the subgraph induced by S contains a perfect matching; motivation for this problem comes from the interest in finding a small number of locations to place pairs of mutually visible guards so that the entire set of guards monitors a given area. The paired-domination problem on general graphs is known to be NP-complete.In this paper, we consider the paired-domination problem on permutation graphs. We define an embedding of permutation graphs in the plane which enables us to obtain an equivalent version of the problem involving points in the plane, and we describe a sweeping algorithm for this problem; given the permutation over the set N n = {1, 2, . . . , n} defining a permutation graph on n vertices, our algorithm computes a paired-dominating set of the graph in O(n) time, and is therefore optimal.

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Domination Problems

Shen¹

2011

Wiley Encyclopedia of Operations Research and Management Science

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In this article, we focus on dominating set problems in graphs. We review the literature that analyzes domination from different aspects, including bounds for the domination number, the algorithmic complexity, and valid inequality chains associated with each domination variant. In particular, we discuss the relationship between domination, independence, and irredundance, and reveal the well‐known domination chain and its properties. As many domination‐related problems can be solved in polynomial time for certain types of graphs (e.g., trees), we summarize start‐of‐the‐art results of polynomial algorithms regarding the domination‐related parameters of graphs. We also describe applications such as wireless ad hoc networks, social networks, and broadcast domination in which domination or its variants can be suitably implemented, as well as associated approximation algorithms, in order to emphasize the importance of the domination in graphs from a practical perspective.

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Paired-domination of Cartesian products of graphs and rainbow domination

Cited by 40 publications

References 5 publications

Vizing's conjecture: a survey and recent results

Vizing's conjecture: a survey and recent results

An O(n)-Time Algorithm for the Paired-Domination Problem on Permutation Graphs

Domination Problems

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