Papers › Critical (P₅,dart)-Free Graphs
Critical (P₅,dart)-Free Graphs
Wen Xia, Jorik Jooken, Jan Goedgebeur, Shenwei Huang
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Given two graphs H₁ and H₂, a graph is (H₁,H₂)-free if it contains no induced subgraph isomorphic to H₁ nor H₂. Let Pₜ be the path on t vertices. 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 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.
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