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.