Contractible Subgraphs in 3-Connected Graphs

A subgraph H of a 3-connected finite graph G is called contractible if H is connected and G−V(H) is 2-connected. This work is concerned with a conjecture of McCuaig and Ota which states that for any given k there exists an f(k) such that any 3-connected graph on at least f(k) vertices possesses a contractible subgraph on k vertices. We prove this for k⩽4 and consider restrictions to maximal planar graphs, Halin graphs, line graphs of 6-edge-connected graphs, 5-connected graphs of bounded degree, and AT-free graphs.