Discussiones Mathematicae Graph Theory 27(2) (2007) 313-321
doi: 10.7151/dmgt.1363

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ON Θ-GRAPHS OF PARTIAL CUBES

Sandi Klavžar

Department of Mathematics and Computer Science
FNM, University of Maribor
Gosposvetska 84, 2000 Maribor, Slovenia
e-mail: sandi.klavzar@uni-mb.si

Matjaz Kovse

Institute of Mathematics, Physics and Mechanics
Gosposvetska 84, 2000 Maribor, Slovenia
e-mail: matjaz.kovse@uni-mb.si

Abstract

The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K1 by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ-graphs.

Keywords: intersection graph, partial cube, median graph, expansion theorem, Cartesian product of graphs.

2000 Mathematics Subject Classification: 05C75, 05C12.

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Received 25 April 2006
Revised 28 December 2006
Accepted 28 December 2006