Counterexample to a conjecture on the structure of bipartite partitionable graphs

• Autorzy: Richard G. Gibson , Christina M. Mynhardt
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

domination; prism fixer; symmetric dominating set; bipartite graph

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2007
• Tom: 27
• Numer: 3
• Strony: 527-540

Daty

wydano

2007

otrzymano

2006-08-21

poprawiono

2007-02-21

zaakceptowano

2007-03-07

Twórcy

autor

Richard G. Gibson

• Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, BC Canada V8W 3P4

autor

Christina M. Mynhardt

• Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, BC Canada V8W 3P4

Bibliografia

• [1] D.W. Bange, A.E. Barkauskas and P.J. Slater, Efficient dominating sets in graphs, in: R.D. Ringeisen and F.S. Roberts, eds, Applications of Discrete Mathematics 189-199 (SIAM, Philadelphia, PA, 1988).
• [2] A.P. Burger, C.M. Mynhardt and W.D. Weakley, On the domination number of prisms of graphs, Discuss. Math. Graph Theory 24 (2004) 303-318, doi: 10.7151/dmgt.1233.
• [3] G. Chartrand and L. Leśniak, Graphs and Digraphs, Third Edition (Chapman & Hall, London, 1996).
• [4] B.L. Hartnell and D.F. Rall, On Vizing's conjecture, Congr. Numer. 82 (1991) 87-96.
• [5] B.L. Hartnell and D.F. Rall, On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24 (2004) 389-402, doi: 10.7151/dmgt.1238.
• [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
• [7] C.M. Mynhardt and Zhixia Xu, Domination in prisms of graphs: Universal fixers, Utilitas Math., to appear.
• [8] P.R.J. Östergå rd and W.D. Weakley, Classification of binary covering codes, J. Combin. Des. 8 (2000) 391-401, doi: 10.1002/1520-6610(2000)8:6<391::AID-JCD1>3.0.CO;2-R
• [9] M. Schurch, Domination Parameters for Prisms of Graphs (Master's thesis, University of Victoria, 2005).
• [10] C.B. Smart and P.J. Slater, Complexity results for closed neighborhood order parameters, Congr. Numer. 112 (1995) 83-96.
• [11] V.G. Vizing, Some unsolved problems in graph theory, Uspehi Mat. Nauk 23 (1968) 117-134.

Identyfikatory

DOI

10.7151/dmgt.1377