On median graphs and median grid graphs

On Median Graphs and Median Grid Graphs on Median Graphs and Median Grid Graphs

Let e be an edge of a median graph G which is contained in exactly one 4-cycle. Then it is proved that G n e is a median graph. The converse holds as well provided that e is not contained is a subgraph isomorphic to the 3-cube minus a vertex. These results are used to give several characterizations of median grid graphs. (Grid graphs are subgraphs of complete grids, i.e. of the Cartesian produc...

On median graphs and median grid graphs

Let G be a Q4-free median graph on n vertices and m edges. Let k be the number of equivalence classes of Djoković-Winkler’s relation Θ and let h be the number of Q3’s in G. Then we prove that 2n −m − k + h = 2. We also characterize median grid graphs in several different ways, for instance, they are the grid graphs with m− n + 1 squares. To obtain these results we introduce the notion of square...

The p-median and p-center Problems on Bipartite Graphs

Let $G$ be a bipartite graph. In this paper we consider the two kind of location problems namely $p$-center and $p$-median problems on bipartite graphs. The $p$-center and $p$-median problems asks to find a subset of vertices of cardinality $p$, so that respectively the maximum and sum of the distances from this set to all other vertices in $G$ is minimized. For each case we present some proper...

Relations between Median Graphs, Semi-median Graphs and Partial Cubes

A rich structure theory has been developed in the last three decades for graphs embed dable into hypercubes in particular for isometric subgraphs of hypercubes also known as partial cubes and for median graphs Median graphs which constitute a proper subclass of partial cubes have attracted considerable attention In order to better understand them semi median graphs have been introduced but have...

On cube-free median graphs