The decay number and the maximum genus of diameter 2 graphs Original Research Article

Let ζ(G) (resp. ξ(G)) be the minimum number of components (resp. odd size components) of a co-tree of a connected graph G. For every 2-connected graph G of diameter 2, it is known that m(G)⩾2n(G)−5 and ξ(G)⩽ζ(G)⩽4. These results define three classes of extremal graphs. In this paper, we prove that they are the same, with the exception of loops added to vertices.