Resolving Restrained Domination in Graphs

Monsanto, Gerald Bacon; Rara, Helen M.

doi:10.29020/nybg.ejpam.v14i3.3985

Cited by 14 publications

(17 citation statements)

References 4 publications

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“…It's noteworthy that the set 𝑆 = 𝑉(𝐺) constitutes a restrained dominating set, and determining is 𝛾 𝑟 (𝐺), an NP-complete decision problem [19]. Several studies on restrained domination in graphs can be found in papers [20][21][22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Outer-restrained Domination in the Lexicographic Product of Two Graphs

et al. 2024

IJFMR

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Let G be a connected simple graph. A set S⊆V(G) is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in (G)∖S . A set S of vertices of a graph G is an outer-restrained dominating set if every vertex not in S is adjacent to some vertex in S and V(G)∖S is a restrained set. The outer-restrained domination number of G, denoted by (γ_r ) ̃(G) is the minimum cardinality of an outer-restrained dominating set of G. An outer-restrained set of cardinality (γ_r ) ̃(G) will be called a (γ_r ) ̃(G) -set. This study is an extension of an existing research on outer-restrained domination in graphs. In this paper, we characterized the outer-restrained domination in graphs under the lexicographic product of two graphs.

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Section: Introductionmentioning

confidence: 99%

Outer-restrained Domination in the Lexicographic Product of Two Graphs

et al. 2024

IJFMR

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“…The dominating set 𝑆 ⊆ 𝑉(𝐺) ∖ 𝐷 is called an inverse dominating set of 𝐺 with respect to a minimum dominating set 𝐷. The concept of inverse domination in graphs was first introduced by Kulli [17], with further information in [18][19][20][21][22][23][24][25][26]. Another type is the disjoint dominating set, defined by Hedetniemi et al [27].…”

Section: Introductionmentioning

confidence: 99%

Disjoint Perfect Secure Domination in the Cartesian Product and Lexicographic Product of Graphs

et al. 2024

IJFMR

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Let G be a graph. A dominating set D⊆V(G) is called a secure dominating set of G if for each vertex u∈V(G)∖D, there exists a vertex v∈D such that uv∈ E(G) and the set (D∖{v})∪{u} is a dominating set of G. If every u∈V(G)∖D is adjacent to exactly one vertex in D, then D is a perfect secure dominating set of G. Let D be a minimum perfect secure dominating set of G. If S⊆V(G)∖D is a perfect secure dominating set of G, then S is called an inverse perfect secure dominating set of G with respect to D. A disjoint perfect secure dominating set of G is the set C=D∪S⊆V(G). Furthermore, the disjoint perfect secure domination number, denoted by γ_ps γ_ps (G), is the minimum cardinality of a disjoint perfect secure dominating set of G. A disjoint perfect secure dominating set of cardinality γ_ps γ_ps (G) is called γ_ps γ_ps-set. In this paper, we extended the study on the concept of disjoint perfect secure domination in graphs. Furthermore, we characterized the disjoint perfect secure domination in the Cartesian product and lexicographic product of two graphs.

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“…Resolving sets was studied by Slater in [12]. Variations of resolving sets and resolving dominating sets were studied in [2,6,7]. The strong resolving sets and strong metric dimension were introduced and characterized by Oellermann and Peters-Fransen [9].…”

Section: Introductionmentioning

confidence: 99%

Strong Resolving Hop Domination in Graphs

Mohamad

Rara²

2023

Eur. J. Pure Appl. Math.

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A vertex w in a connected graph G strongly resolves two distinct vertices u and v in V (G) if v is in any shortest u-w path or if u is in any shortest v-w path. A set W of vertices in G is a strong resolving set G if every two vertices of G are strongly resolved by some vertex of W. A set S subset of V (G) is a strong resolving hop dominating set of G if S is a strong resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the strong resolving hop domination number of G. This paper presents the characterization of the strong resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding strong resolving hop domination number.

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Resolving Restrained Domination in Graphs

Cited by 14 publications

References 4 publications

Outer-restrained Domination in the Lexicographic Product of Two Graphs

Outer-restrained Domination in the Lexicographic Product of Two Graphs

Disjoint Perfect Secure Domination in the Cartesian Product and Lexicographic Product of Graphs

Strong Resolving Hop Domination in Graphs

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