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Total domination in 2-connected graphs and in graphs with no induced 6-cycles

✍ Scribed by Michael A. Henning; Anders Yeo

Publisher: John Wiley and Sons
Year: 2009
Tongue: English
Weight: 272 KB
Volume: 60
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20345

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✦ Synopsis

Abstract

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ~t~(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ~t~(G) ≤ 4__n__/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4__n__/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ~t~(G) ≤ 6__n__/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ~t~(G) ≤ 6__n__/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009

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