Hadwiger number of graphs with small chordalityPetr A. Golovach, Pinar Heggernes, Pim van 't Hof and Christophe Paul
A preliminary version of this paper will appear in the proceedings of WG 2014, the 40th International Workshop on Graph-Theoretic Concepts in Computer Science (held on June 25-27, 2014 in Le Domaine de Chalès, France), Lecture Notes in Computer Science, vol. 8747, pp. 201-213, 2014.
The Hadwiger number of a graph G is the largest integer h such that G has the complete graph Kh 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.