Discussiones Mathematicae Graph Theory 28(2) (2008) 267-284
doi: 10.7151/dmgt.1405

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SECURE DOMINATION AND SECURE TOTAL DOMINATION IN GRAPHS

William F. Klostermeyer

School of Computing
University of North Florida
Jacksonville, FL 32224-2669
e-mail: wkloster@unf.edu

Christina M. Mynhardt

Department of Mathematics and Statistics
University of Victoria, P.O. Box 3060 STN CSC
Victoria, BC, Canada V8W 3R4
e-mail: mynhardt@math.uvic.ca

Abstract

A secure (total) dominating set of a graph G = (V,E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V−X, there exists x ∈ X adjacent to u such that (X−{x})∪{u} is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number γs(G) (γst(G)). We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then γst(G) ≤ γs(G). We also show that γst(G) is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.

Keywords: secure domination, total domination, secure total domination, clique covering.

2000 Mathematics Subject Classification: 05C69.

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Received 29 June 2007
Revised 25 April 2008
Accepted 25 April 2008