Characterization of $P_3$-connected graphs

TL;DR

This work characterizes $P_3$-connected graphs by showing that a simple graph is $P_3$-connected if and only if it is connected and has no non-stable homogeneous set. The proof combines a decomposition into edge-maximal $P_3$-connected subgraphs with a refined neighborhood partition of a chosen vertex and a directed-graph argument to force a single spanning component. A key corollary is that every connected triangle-free graph is $P_3$-connected, highlighting the structural constraints that ensure $P_3$-connectivity. The results build on the framework introduced by Chudnovsky et al. and have implications for related problems such as cop-number bounds in graphs excluding certain induced subgraphs like $P_5$.

Abstract

For any pair of edges $e,f$ of a graph $G$, we say that {\em $e,f$ are $P_3$-connected in $G$} if there exists a sequence of edges $e=e_0,e_1,\ldots, e_k=f$ such that $e_i$ and $e_{i+1}$ are two edges of an induced $3$-vertex path in $G$ for every $0\leq i\leq k-1$. If every pair of edges of $G$ are $P_3$-connected in $G$, then $G$ is {\em $P_3$-connected}. $P_3$-connectivity was first defined by Chudnovsky et al. in 2024 to prove that every connected graph not containing $P_5$ as an induced subgraph has cop number at most two. In this paper, we give a characterization of $P_3$-connected graphs and prove that a simple graph is $P_3$-connected if and only if it is connected and has no homogeneous set whose induced subgraph contains an edge.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof