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–Winklerʹs relation Θ and let h be the number of Q3ʹs 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-edges, i.e. edges contained in exactly one 4-cycle of a graph.