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 \(s\ge 2\), but is NP-hard on chordal graphs for every fixed \(s\ge 2\).
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