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Discussiones Mathematicae Graph Theory 23(2) (2003) 215-225
DOI: https://doi.org/10.7151/dmgt.1198
ARBOREAL STRUCTURE AND REGULAR GRAPHS OF MEDIAN-LIKE CLASSES
Bostjan Bresar1
University of Maribor
FERI, Smetanova 17, 2000 Maribor, Slovenia
e-mail: Bostjan.Bresar@uni-mb.si
Abstract
We consider classes of graphs that enjoy the following properties: they are closed for gated subgraphs, gated amalgamation and Cartesian products, and for any gated subgraph the inverse of the gate function maps vertices to gated subsets. We prove that any graph of such a class contains a peripheral subgraph which is a Cartesian product of two graphs: a gated subgraph of the graph and a prime graph minus a vertex. Therefore, these graphs admit a peripheral elimination procedure which is a generalization of analogous procedure in median graphs. We characterize regular graphs of these classes whenever they enjoy an additional property. As a corollary we derive that regular weakly median graphs are precisely Cartesian products in which each factor is a complete graph or a hyperoctahedron.Keywords: median graph, tree, gatedness, amalgam, periphery, regular graph.
2000 Mathematics Subject Classification: 05C12, 05C75.
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Received 26 September 2001
Revised 6 February 2002
Footnotes:
1Supported by the Ministry of Education, Science and Sport of Slovenia under the grant Z1-3073-0101-01.