## Hadwiger number of graphs with small chordality

*Petr A. Golovach, Pinar Heggernes, Pim van 't Hof and Christophe Paul*

*SIAM Journal on Discrete Mathematics*, vol. 29, no. 3, pp. 1427-1451, 2015.

[__DOI__]
[__Preprint__]

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.

[__DOI__]

### 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 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.