Contractible Subgraphs of Contraction Critically Quasi $5$-Connected Graphs

TL;DR

This paper investigates specific subgraph structures within contraction critically quasi 5-connected graphs, identifying conditions under which small quasi 5-contractible subgraphs exist, advancing understanding of their connectivity properties.

Contribution

It establishes the existence of small quasi 5-contractible subgraphs in certain contraction critically quasi 5-connected graphs under specified local conditions.

Findings

Existence of small quasi 5-contractible subgraphs under given conditions

Characterization of neighborhoods leading to contractible subgraphs

Extension of connectivity theory in quasi 5-connected graphs

Abstract

Let $G$ be a contraction critically quasi $5$-connected graph on at least $14$ vertices. If there is a vertex $x\in V_{4}(G)$ such that $G[N_{G}(x)]\cong K_{1,3}$ or $G[N_{G}(x)]\cong C_{4}$, then $G$ has a quasi $5$-contractible subgraph $H$ such that $0<\|V(H)\|<4$.