Topological containment of the 5-clique minus an edge in 4-connected graphs

The topological containment problem is known to be polynomial-time solvable for any fixed pattern graph H , but good characterisations have been found for only a handful of non-trivial pattern graphs. The complete graph on five vertices, K5, is one pattern graph for which a characterisation has not been found. The discovery of such a characterisation would be of particular interest, due to the Hajós Conjecture. One step towards this may be to find a good characterisation of graphs that do not topologically contain the simpler pattern graph K− 5 , obtained by removing a single edge from K5. This paper makes progress towards achieving this, by showing that every 4-connected graph must contain a K− 5 -subdivision.