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.
Categories: math.CO