Papers › Critical (P₅,W₄)-Free Graphs
Critical (P₅,W₄)-Free Graphs
Wen Xia, Jorik Jooken, Jan Goedgebeur, Iain Beaton, Ben Cameron, Shenwei Huang
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A graph G is k-vertex-critical if χ(G) = k but χ(G-v)<k for all v ∈V(G). A graph is (H₁,H₂)-free if it contains no induced subgraph isomorphic to H₁ nor H₂. A W₄ is the graph consisting of a C₄ plus an additional vertex adjacent to all the vertices of the C₄. We show that there are finitely many k-vertex-critical (P₅,W₄)-free graphs for all k ≥1 and we characterize all $5$-vertex-critical (P₅,W₄)-free graphs. Our results imply the existence of a polynomial-time certifying algorithm to decide the k-colorability of (P₅,W₄)-free graphs for each k ≥1 where the certificate is either a k-coloring or a (k+1)-vertex-critical induced subgraph.
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