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

## TL;DR

This paper investigates the topological containment problem in graph theory, focusing on 4-connected graphs and their subdivisions, and makes progress towards characterising graphs that do not contain a specific pattern related to the Hajós Conjecture.

## Contribution

It proves that every 4-connected graph must contain a subdivision of the graph obtained by removing an edge from K_5, advancing understanding of topological containment for this pattern.

## Key findings

- Every 4-connected graph contains a K_5^- subdivision.
- Progress towards characterising graphs avoiding K_5^- topological containment.
- Supports the pursuit of characterisations related to the Hajós Conjecture.

## Abstract

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, $K_5$, 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 Haj\'os 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 $K_5$. This paper makes progress towards achieving this, by showing that every 4-connected graph must contain a $K_5^-$-subdivision.