Median Graphs and Triangle-Free Graphs

Let M(m,n) be the complexity of checking whether a graph G with m edges and n vertices is a median graph. We show that the complexity of checking whether G is triangle-free is at most M(m,m). Conversely, we prove that the complexity of checking whether a given graph is a median graph is at most O(m log n) + T (m log n, n), where T (m,n) is the complexity of finding all triangles of the graph. We also demonstrate that, intuitively speaking, there are as many median graphs as there are triangle-free graphs. Finally, these results enable us to prove that the complexity of recognizing planar median graphs is linear.