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Discussiones Mathematicae Graph Theory 26(3) (2006) 457-474
DOI: https://doi.org/10.7151/dmgt.1338

SOME RECENT RESULTS ON DOMINATION IN GRAPHS

Michael D. Plummer

Department of Mathematics
Vanderbilt University
Nashville, Tennessee 37240, USA
e-mail: michael.d.plummer@vanderbilt.edu

Abstract

In this paper, we survey some new results in four areas of domination in graphs, namely:

(1) the toughness and matching structure of graphs having domination number 3 and which are "critical" in the sense that if one adds any missing edge, the domination number falls to 2;

(2) the matching structure of graphs having domination number 3 and which are "critical" in the sense that if one deletes any vertex, the domination number falls to 2;

(3) upper bounds on the domination number of cubic graphs; and

(4) upper bounds on the domination number of graphs embedded in surfaces.

Keywords: domination, matching, toughness, cubic graph, triangulation, genus.

2000 Mathematics Subject Classification: 05C10, 05C69, 05C70.

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Received 23 November 2005