Paired Domination in Graphs

“…Paired-domination was introduced by Haynes and Slater [1] and is studied, for example, in [2][3][4][5][6][7]. For more details on paired-domination, we refer the reader to the recent book chapter [8].…”

Section: Introductionmentioning

confidence: 99%

A Note on the Paired-Domination Subdivision Number of Trees

et al. 2021

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“…With respect to the matching M , two vertices joined by an edge of M are paired and are called partners in S. The paired domination number, γ pr (G), of G is the minimum cardinality of a PD-set of G. We call a PD-set of G of cardinality γ pr (G) a γ pr -set of G. We note that the paired domination number γ pr (G) is an even integer. For a recent survey on paired domination in graphs, we refer the reader to the book chapter [8].…”

Section: Introductionmentioning

confidence: 99%

Paired domination stability in graphs

et al. 2022

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A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, γ pr (G), of G is the minimum cardinality of a paired dominating set of G. A set of vertices whose removal from G produces a graph without isolated vertices is called a non-isolating set. The minimum cardinality of a non-isolating set of vertices whose removal decreases the paired domination number is the γ − pr -stability of G, denoted st − γpr (G). The paired domination stability of G is the minimum cardinality of a non-isolating set of vertices in G whose removal changes the paired domination number. We establish properties of paired domination stability in graphs. We prove that if G is a connected graph with γ pr (G) ≥ 4, then st − γpr (G) ≤ 2∆(G) where ∆(G) is the maximum degree in G, and we characterize the infinite family of trees that achieve equality in this upper bound.

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“…Extended forms of domination in graphs have been vast in the domain literature. Some very recent forms include broadcast domination [18], pitchfork domination [19], Roman domination [20], double Roman domination [21], triple Roman domination [22], captive domination [23], outer-convex domination [24], and paired domination [25], among others. The trajectory of these topics has been exponential in the last decade.…”

Section: Introductionmentioning

confidence: 99%

Domination in Fuzzy Directed Graphs

et al. 2021

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A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let GD*=V,A be a directed simple graph, where V is a finite nonempty set and A=x,y:x,y∈V,x≠y. A fuzzy digraph GD=σD,μD is a pair of two functions σD:V→0,1 and μD:A→0,1, such that μDx,y≤σDx∧σDy, where x,y∈V. An edge μDx,y of a fuzzy digraph is called an effective edge if μDx,y=σDx∧σDy. Let x,y∈V. The vertex σDx dominates σDy in GD if μDx,y is an effective edge. Let S⊆V, u∈V\S, and v∈S. A subset σDS⊆σD is a dominating set of GD if, for every σDu∈σD\σDS, there exists σDv∈σDS, such that σDv dominates σDu. The minimum dominating set of a fuzzy digraph GD is called the domination number of a fuzzy digraph and is denoted by γGD. In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.

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Paired Domination in Graphs

Cited by 14 publications

References 53 publications

A Note on the Paired-Domination Subdivision Number of Trees

A Note on the Paired-Domination Subdivision Number of Trees

Paired domination stability in graphs

Domination in Fuzzy Directed Graphs

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