Hadwiger number of graphs with small chordality

TL;DR

This paper investigates the computational complexity of determining the Hadwiger number in various graph classes, showing NP-hardness in some cases and polynomial-time algorithms in others, and extends the problem to diameter-constrained minors.

Contribution

It establishes NP-hardness of the Hadwiger number problem on co-bipartite graphs and chordal graphs, and provides polynomial algorithms for cographs, bipartite permutation graphs, and AT-free graphs.

Findings

NP-hard on co-bipartite graphs

Polynomial-time on cographs and bipartite permutation graphs

Polynomial-time on AT-free graphs for s>=2

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.