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 turned out to form a rather interesting class of graphs by themselves This class lies strictly between median graphs and partial cubes and satis es an expansion theorem for which we give a self contained proof Furthermore we show that these graphs can be characterized as tiled partial cubes and we prove that for a semi median graph G with n vertices m edges and k equivalence classes of Djokovi c s relation we have n m k Moreover equality holds if and only if G contains no K C t with t as a subgraph For median graphs we show that they can be characterized as semi median graphs which contain no convex Q i e the cube minus a vertex as a convex subgraph Also we introduce the concept of weak convexity and use it among other things to prove that median graphs are bipartite meshed graphs with weakly convex intervals Supported by the Ministry of Science and Technology of Slovenia under the grant J and by the SWON the Netherlands Supported by the Ministry of Science and Technology of Slovenia and by the NUFFIC the Netherlands Supported by the Ministry of Science and Technology of Slovenia under the grant J