Domination versus disjunctive domination in trees

Henning, Michael A.; Marcon, Sinclair Antony

doi:10.1016/j.dam.2014.10.036

Cited by 15 publications

(10 citation statements)

References 3 publications

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“…Observation 2.1 [5] If T is a tree of order at least 3, then we can choose a γ d 2 -set of T contains no leaf.…”

Section: Resultsmentioning

confidence: 99%

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Disjunctive domination in trees

Zhuang¹

2018

Preprint

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In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the domination number. Given the sheer scale of modern networks, many existing domination type structures are expensive to implement. Variations on the theme of dominating sets studied to date tend to focus on adding restrictions which in turn raises their implementation costs. As an alternative route a relaxation of the domination number, called disjunctive domination, was proposed and studied by Goddard et al. A set D of vertices in G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γ d 2 (G), of G is the minimum cardinality of a disjunctive dominating set in G. We show that if T is a tree of order n with l leaves and s support vertices, then n−l+3 4 ≤ γ d 2 (T ) ≤ n+l+s 4 . Moreover, we characterize the families of trees which attain these bounds.

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“…Observation 2.1 [5] If T is a tree of order at least 3, then we can choose a γ d 2 -set of T contains no leaf.…”

Section: Resultsmentioning

confidence: 99%

“…Every dominating set is a 2DD-set. The concept of disjunctive domination in graphs has been studied in [2][3][4][5] and elsewhere.…”

Section: Introductionmentioning

confidence: 99%

Disjunctive domination in trees

Zhuang¹

2018

Preprint

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“…The concept of disjunctive domination has been introduced recently in 2014 [8] and further studied in [11]. In [8], Goddard et al have proven bounds on disjunctive domination number for specially regular graphs and claw-free graphs.…”

Section: Disjunctive Domination Decision Problem (Dddp)mentioning

confidence: 99%

Algorithmic Aspects of Disjunctive Domination in Graphs

Panda

Pandey

Shrivastava

2015

Lecture Notes in Computer Science

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For a graph G = (V, E), a set D ⊆ V is called a disjunctive dominating set of G if for every vertex v ∈ V \ D, v is either adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The cardinality of a minimum disjunctive dominating set of G is called the disjunctive domination number of graph G, and is denoted by γ. Given a positive integer k and a graph G, the DISJUNCTIVE DOMINATION DECISION PROBLEM (DDDP) is to decide whether G has a disjunctive dominating set of cardinality at most k. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a (ln(∆ 2 + ∆ + 2) + 1)-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within (1 − ) ln(|V |) for any > 0 unless NP ⊆ DTIME(|V | O(log log |V |) ). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 3.

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“…Every dominating set is a 2DD-set. The concept of disjunctive domination in graphs has been studied in [3,[7][8][9][10][11][12] and elsewhere.…”

Section: Introductionmentioning

confidence: 99%

Disjunctive domination in graphs with minimum degree at least two

Zhuang

2021

Preprint

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A set D of vertices in G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γ d 2 (G), of G is the minimum cardinality of a disjunctive dominating set in G. In this paper, we show that if G be a graph of order at least 3, δ(G) ≥ 2 and with no component isomorphic to any of eight forbidden graphs, then γ d 2 (G) ≤ |G| 3 . Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also study the case that G is a claw-free graph with minimum degree at least two.

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Domination versus disjunctive domination in trees

Cited by 15 publications

References 3 publications

Disjunctive domination in trees

Disjunctive domination in trees

Algorithmic Aspects of Disjunctive Domination in Graphs

Disjunctive domination in graphs with minimum degree at least two

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