A forbidden pair for quasi 5-contractible edges

Abstract

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. If every quasi $k$-connected graph without a quasi $k$-contractible edge has either $H_{1}$ or $H_{2}$ as a subgraph, then an unordered pair of graphs $\{H_{1}, H_{2}\}$ is said to be a forbidden pair for quasi $k$-contractible edges. We prove that $\{K_{4}^{-}, \overline{P_{5}}\}$ is a forbidden pair for quasi 5-contractible edges, where $K_{4}^{-}$ is the graph obtained from $K_{4}$ by removing just one edge and $\overline{P_{5}}$ is the complement of a path on five vertices.