Some properties of minimally nonperfectly divisible graphs

TL;DR

This paper explores properties of minimally nonperfectly divisible graphs, examining their relationship with perfect divisibility, and shows that certain classes of these graphs lack clique cutsets, answering a question in graph theory.

Contribution

It establishes the relationship between perfect divisibility and perfect weighted divisibility, and proves that specific minimally nonperfectly divisible graphs contain no clique cutset.

Findings

2P3-free minimally nonperfectly divisible graphs contain no clique cutset

Claw-free minimally nonperfectly divisible graphs contain no clique cutset

Provides a conditional answer to a question by Hoàng

Abstract

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$, and a graph $G$ is perfectly weight divisible if for every positive integral weight function on $V(G)$ and each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. A clique $X$ of a connected graph $G$ is called a clique cutset if $G-X$ is disconnected. In this paper, we investigate the relationship between the perfect divisibility of a graph and its perfect weighted divisibility. We also show that $2P_3$-free or claw-free minimally nonperfectly divisible graphs contain no clique cutset, that conditionally answers a question of Ho\`ang [Discrete Math. \textbf{349} (2025) 114809].