Hadwiger Number of Graphs with Small Chordality

The Hadwiger number of a graph G is the largest integer h such that G has the complete graph K-h as a minor. We show that the problem of determining the Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h such that G has a minor with h vertices and diameter at most s. We show that this problem can be solved in polynomial time on AT-free graphs when s >= 2, but is NP-hard on chordal graphs for every fixed s >= 2.