Upper total domination in claw-free cubic graphs
type: Item
instance of: scholarly article
title: Upper total domination in claw-free cubic graphs
zbMATH Open document ID: 1498.05200
author: Ammar Babikir
author: Michael A. Henning
published in: Graphs and Combinatorics
publication date: 2022-10-13T00:00:00Z
review text: Let \(G = (V, E)\) be a simple graph of order \(n\). A total dominating set of \(G\) is a subset \(D\) of \(V\) such that every vertex of \(V\) is adjacent to some vertices of \(D\). The total domination number of \(G\) is equal to the minimum cardinality of a total dominating set in \(G\) and is denoted by \(\gamma_t(G)\). The upper total domination number, \(\Gamma_t(G)\), of \(G\) is the maximum cardinality of a minimal total dominating set of \(G\). A claw-free graph is a graph that does not contain a claw \(K_{1,3}\) as an induced subgraph. Let \(G\) be a connected, claw-free, cubic graph of order \(n\). Since \(\gamma_t(G)\geq \frac{n}{\Delta(G)}\) for every isolate-free graph, we observe that \(\gamma_t(G)\geq \frac{1}{3}n\), and this bound is sharp. Recently, it is proved that if we exclude four graphs, then \(\gamma_t(G)\leq \frac{3}{7}n\). In this paper, the authors show that \(\frac{4}{9}n \leq \Gamma_t(G)\leq \frac{3}{5}n\).
reviewed by: Saeid Alikhani
Mathematics Subject Classification ID: 05C69
zbMATH DE Number: 7601301
zbMATH Keywords: total domination
zbMATH Keywords: upper total domination
zbMATH Keywords: claw-free cubic graph
MaRDI profile type: Publication
full work available at URL: s00373-022-02581-0
OpenAlex ID: W4304692625
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DOI: 10.1007/S00373-022-02581-0
description: scientific article; zbMATH DE number 7601301