The Median Problem on Symmetric Bipartite Graphs

Abstract

In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. A bipartite graph G is symmetric if for a bi-partition (X, Y) of G, there is a map f from X onto Y such that if $(u,f(v)) \in E(G)$, then $(v,f(u)) \in E(G)$, where $u,v\in X$. In this paper we show, by construction, that any symmetric bipartite graph is a median (anti-median, center) of another symmetric bipartite graph. We also obtain results on median and anti-median problem on square graphs of bi-partite graphs with equal partitions.