Critical (P5,dart)-free graphs

• Published in: Discrete Applied Mathematics Vol. 366; pp. 44 - 52
• Main Authors: Xia, Wen, Jooken, Jorik, Goedgebeur, Jan, Huang, Shenwei
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.05.2025
• Subjects: Graph coloring; k-critical graphs; Polynomial-time algorithms; Strong perfect graph theorem; Polynomial-time algorithms; k-critical graphs; Graph coloring; Strong perfect graph theorem
• ISSN: 0166-218X
• DOI: 10.1016/j.dam.2025.01.011

Given two graphs H1 and H2, a graph is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 nor H2. A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex with degree 3 in the diamond. In this paper, we show that there are finitely many k-vertex-critical (P5,dart)-free graphs for k≥1. To prove these results, we use induction on k and perform a careful structural analysis via the Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Moreover, for k∈{5,6,7} we characterize all k-vertex-critical (P5,dart)-free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the k-colorability of (P5,dart)-free graphs for k≥1 where the certificate is either a k-coloring or a (k+1)-vertex-critical induced subgraph.