Elsevier

Discrete Mathematics

Volume 313, Issue 2, 28 January 2013, Pages 174-181
Discrete Mathematics

Total domination and matching numbers in graphs with all vertices in triangles

https://doi.org/10.1016/j.disc.2012.09.024Get rights and content
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Abstract

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. The matching number of G is the maximum cardinality of a matching of G. A set S of vertices in G is a total dominating set 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 of G. We prove that if all vertices of G belong to a triangle, then the total domination number of G is bounded above by its matching number. We in fact prove a slightly stronger result and as a consequence of this stronger result, we prove a Graffiti conjecture that relates the total domination and matching numbers in a graph.

Keywords

Matching number
Total domination number

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