A probabilistic algorithm for bounding the total restrained domination number of a K1,ℓ -free graph

Ernst J. Joubert

doi:10.1016/j.dam.2024.07.009

A probabilistic algorithm for bounding the total restrained domination number of a K_1,ℓ -free graph

Ernst J. Joubert

Mathematics and Applied Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S, and every vertex in V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted γ_tr(G), is the smallest cardinality of a total restrained dominating set of G. In this paper we show that if G is a K_1,ℓ-free graph with δ≥ℓ≥3 and δ≥5, then γ_tr(G)≤n1−[Formula presented]+o_δ(1).

Original language	English
Pages (from-to)	429-439
Number of pages	11
Journal	Discrete Applied Mathematics
Volume	357
DOIs	https://doi.org/10.1016/j.dam.2024.07.009
Publication status	Published - 15 Nov 2024

Keywords

Domination
Graph
Total restrained domination
Upper bound

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.1016/j.dam.2024.07.009

Cite this

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title = "A probabilistic algorithm for bounding the total restrained domination number of a K1,ℓ -free graph",

abstract = "Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S, and every vertex in V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted γtr(G), is the smallest cardinality of a total restrained dominating set of G. In this paper we show that if G is a K1,ℓ-free graph with δ≥ℓ≥3 and δ≥5, then γtr(G)≤n1−[Formula presented]+oδ(1).",

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N2 - Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S, and every vertex in V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted γtr(G), is the smallest cardinality of a total restrained dominating set of G. In this paper we show that if G is a K1,ℓ-free graph with δ≥ℓ≥3 and δ≥5, then γtr(G)≤n1−[Formula presented]+oδ(1).

AB - Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S, and every vertex in V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted γtr(G), is the smallest cardinality of a total restrained dominating set of G. In this paper we show that if G is a K1,ℓ-free graph with δ≥ℓ≥3 and δ≥5, then γtr(G)≤n1−[Formula presented]+oδ(1).

KW - Domination

KW - Graph

KW - Total restrained domination

KW - Upper bound

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