Contractible edges in 3-connected graphs that preserve a minor

Abstract

Let G be a 3-connected graph with a 3-connected (or sufficiently small) simple minor H. We establish that G has a forest F with at least ⌈(|G|−|H|+1)/2⌉ edges such that G/e is 3-connected with an H-minor for each e∈E(F). Moreover, we may pick F with |G|−|H| edges provided G is triangle-free. These results are sharp. Our result generalizes a previous one by Ando et al., which establishes that a 3-connected graph G has at least ⌈|G|/2⌉ contractible edges. As another consequence, each triangle-free 3-connected graph has an spanning tree of contractible edges. Our results follow from a more general theorem on graph minors, a splitter theorem, which is also established here.