DMGT

Discussiones Mathematicae Graph Theory 25(1-2) (2005) 183-196
DOI: https://doi.org/10.7151/dmgt.1271

MEDIAN AND QUASI-MEDIAN DIRECT PRODUCTS OF GRAPHS

Bostjan Bresar

University of Maribor, FEECS
Smetanova 17, 2000 Maribor, Slovenia
e-mail: bostjan.bresar@uni-mb.si

Pranava K. Jha

Department of Computer Science
St. Cloud State University
720 Fourth Ave. S., St. Cloud, MN 56301, USA
e-mail: pkjha@stcloudstate.edu

Sandi Klavžar

Department of Mathematics and Computer Science, PEF
University of Maribor
Koroska 160, 2000 Maribor, Slovenia
e-mail: sandi.klavzar@uni-mb.si

Blaz Zmazek

University of Maribor, FME
Smetanova 17, 2000 Maribor, Slovenia
e-mail: blaz.zmazek@uni-mb.si

Abstract

Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G×P₃ is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G×K₂ are proved, and it is shown that the only nonbipartite quasi-median direct product is K₃×K₃.

Keywords: median graph, direct product, quasi-median graph, isometric embeddings, convexity.

2000 Mathematics Subject Classification: 05C75, 05C12.

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Received 29 November 2003
Revised 1 September 2004