Discussiones Mathematicae Graph Theory 30(1) (2010) 17-32
doi: 10.7151/dmgt.1473

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THE PERIPHERY GRAPH OF A MEDIAN GRAPH

Bostjan Bresar

Faculty of Natural Sciences and Mathematics
University of Maribor, Slovenia
e-mail: bostjan.bresar@uni-mb.si

Manoj Changat

Department of Futures Studies
University of Kerala, Trivandrum-695034, India
e-mail: mchangat@gmail.com

Ajitha R. Subhamathi

Department of Futures Studies
University of Kerala, Trivandrum-695034, India
e-mail: ajithars@gmail.com

Aleksandra Tepeh

Faculty of Electrical Engineering and Computer Science
University of Maribor, Slovenia
e-mail: aleksandra.tepeh@uni-mb.si

Abstract

The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join of two graphs and show that they are precisely Cartesian products of median graphs. Path-like median graphs are introduced as the graphs whose periphery graph has independence number 2, and it is proved that there are path-like median graphs with arbitrarily large geodetic number. Peripheral expansion with respect to periphery graph is also considered, and connections with the concept of crossing graph are established.

Keywords: median graph, Cartesian product, geodesic, periphery, peripheral expansion.

2010 Mathematical Subject Classification: 05C12, 05C75.

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Received 11 June 2008
Revised 5 January 2009
Accepted 5 January 2009