Discussiones Mathematicae Graph Theory 28(3) (2008) 453-462
doi: 10.7151/dmgt.1419

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THE BONDAGE NUMBER OF GRAPHS: GOOD AND BAD VERTICES

Vladimir Samodivkin

Department of Mathematics
University of Architecture Civil Engineering and Geodesy
Hristo Smirnenski 1 Blv., 1046 Sofia, Bulgaria
e-mail: vlsam_fte@uacg.bg

Abstract

The domination number γ(G) of a graph G is the minimum number of vertices in a set D such that every vertex of the graph is either in D or is adjacent to a member of D. Any dominating set D of a graph G with |D| = γ(G) is called a γ-set of G. A vertex x of a graph G is called: (i) γ-good if x belongs to some γ-set and (ii) γ-bad if x belongs to no γ-set. The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater then γ(G). In this paper we present new sharp upper bounds for b(G) in terms of γ-good and γ-bad vertices of G.

Keywords: bondage number, γ-bad/good vertex.

2000 Mathematics Subject Classification: 05C69.

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Received 20 September 2007
Revised 14 July 2008
Accepted 14 July 2008