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Counterexample to a conjecture on the structure of bipartite partitionable graphs

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Gibson R., Mynhardt C.

Discussiones Mathematicae Graph Theory

2007

tom 27

nr 3

527-540

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 $V(G)-N[D_i] = D_j$, 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 $K_{2t,2t}$ 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.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Gibson R.

1 Mynhardt C.

1 2007

Wyniki wyszukiwania

Counterexample to a conjecture on the structure of bipartite partitionable graphs