Hadwiger Number of Graphs with Small Chordality

Abstract

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 \sf 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 $s\geq 2$ but is \sf NP-hard on chordal graphs for every fixed $s\geq 2$.