Critical $(P_5,W_4)$-Free Graphs

Abstract

A graph $G$ is $k$-vertex-critical if $\ddot{I}‡(G)=k$ but $\ddot{I}‡(G-v)<k$ for all $v\in V(G)$. A graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A $W_4$ is the graph consisting of a $C_4$ plus an additional vertex adjacent to all the vertices of the $C_4$. We show that there are finitely many $k$-vertex-critical $(P_5,W_4)$-free graphs for all $k\ge 1$ and we characterize all $5$-vertex-critical $(P_5,W_4)$-free graphs. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,W_4)$-free graphs for each $k\ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.

arXiv admin note: text overlap with arXiv:2308.03414, arXiv:2403.05611