## 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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