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- W4385681824 abstract "Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. Let $P_t$ 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 $(P_5,dart)$-free graphs for $k ge 1$ 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 $k in {5, 6, 7}$ we characterize all $k$-vertex-critical $(P_5,dart)$-free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,dart)$-free graphs for $k ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph." @default.
- W4385681824 created "2023-08-09" @default.
- W4385681824 creator A5073546070 @default.
- W4385681824 creator A5076884755 @default.
- W4385681824 date "2023-08-07" @default.
- W4385681824 modified "2023-10-13" @default.
- W4385681824 title "Critical $(P_5,dart)$-Free Graphs" @default.
- W4385681824 doi "https://doi.org/10.48550/arxiv.2308.03414" @default.
- W4385681824 hasPublicationYear "2023" @default.
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