The decay number and the maximum genus of diameter 2 graphs (Q1841903)

Let \(\chi(G)\) (resp. \(\xi(G)\)) be the minimum number of components (resp. of odd size components) of a cotree of a connected graph \(G\). If \(G\) is a 2-connected graph of diameter \(2\), then it is known that \(m(G)\geq 2\cdot n(G)-5\) and \(\xi(G)\leq \chi(G)\leq 4\). These results define three classes of extremal graphs. In this paper the authors prove that they coincide, with the exception of loops added to vertices.