Counterexample to a conjecture on the structure of bipartite partitionable graphs
Richard G. Gibson ; Christina M. Mynhardt
Discussiones Mathematicae Graph Theory, Tome 27 (2007), p. 527-540 / Harvested from The Polish Digital Mathematics Library
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A graph G is called a prism fixer if γ(G×K₂) = γ(G), where γ(G) denotes the domination number of G. A symmetric γ-set of G is a minimum dominating set D which admits a partition D = D₁∪ D₂ such that V(G)-N[Di]=Dj, i,j = 1,2, i ≠ j. It is known that G is a prism fixer if and only if G has a symmetric γ-set. Hartnell and Rall [On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24 (2004), 389-402] conjectured that if G is a connected, bipartite graph such that V(G) can be partitioned into symmetric γ-sets, then G ≅ C₄ or G can be obtained from K2t,2t by removing the edges of t vertex-disjoint 4-cycles. We construct a counterexample to this conjecture and prove an alternative result on the structure of such bipartite graphs.

Publié le : 2007-01-01
EUDML-ID : urn:eudml:doc:270301 
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Richard G. Gibson; Christina M. Mynhardt. Counterexample to a conjecture on the structure of bipartite partitionable graphs. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 527-540. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1377/

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