Abstract
For a median graph G and a vertex v of G that is not a cut-vertex we show that G - v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v . This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whose all vertex-deleted subgraphs are not median (respectively, are median). Moreover, two analogous characterizations for edge-deleted median graphs are given.
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