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On the Finiteness of k-Vertex-Critical \(2P_2\)-Free Graphs with Forbidden Induced Squids or Bulls

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Combinatorial Algorithms (IWOCA 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14764))

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Abstract

A graph G is k-vertex-critical if \(\chi (G)=k\) but \(\chi (G-v)<k\) for all \(v\in V(G)\), where \(\chi (G)\) is the chromatic number of G. A graph is \((H_1,H_2)\)-free if it contains no induced subgraph isomorphic to \(H_1\) or \(H_2\). We show that there are only finitely many k-vertex-critical \((2P_2,H)\)-free graphs for all k when H is isomorphic to any of the following graphs:

  • \((m, \ell )\)-squid for \(m=3,4\) and any \(\ell \ge 1\) (where an \((m,\ell )\)-squid is the graph obtained from an m-cycle by attaching \(\ell \) leaves to a single vertex of the cycle),

  • bull,

  • chair,

  • \(claw+P_1\), or

  • \(\overline{diamond+P_1}\).

The latter three are corollaries of the \((m, \ell )\)-squid results, while the bull is handled on its own. For each of the graphs H as above and any fixed k, our results imply the existence of polynomial-time certifying algorithms for deciding the k-colourability problem for \((2P_2,H)\)-free graphs. Further, our structural classifications allow us to exhaustively generate, with the aid of a computer search, all k-vertex-critical \((2P_2,H)\)-free graphs for \(k\le 7\) when \(H=bull\) or \(H=(4,1)\)-squid (also known as banner).

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Notes

  1. 1.

    Since this paper was submitted, the \(claw+P_1\) and \(\overline{K_3+2P_1}\) cases were both proved in [37].

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Acknowledgments

The first two authors were supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) USRA program. The third author also gratefully acknowledges research support from NSERC (grants RGPIN-2022-03697 and DGECR-2022-00446) and Alberta Innovates. The research support from Alberta Innovates was used by the third author to hire the fourth author.

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Authors and Affiliations

  1. The King’s University, Edmonton, AB, Canada

    Melvin Adekanye, Christopher Bury & Thaler Knodel

  2. University of Prince Edward Island, Charlottetown, PE, Canada

    Ben Cameron

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  1. Melvin Adekanye
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Correspondence to Ben Cameron .

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  1. University of Salerno, Fisciano, Italy

    Adele Anna Rescigno

  2. University of Salerno, Fisciano, Salerno, Italy

    Ugo Vaccaro

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Adekanye, M., Bury, C., Cameron, B., Knodel, T. (2024). On the Finiteness of k-Vertex-Critical \(2P_2\)-Free Graphs with Forbidden Induced Squids or Bulls. In: Rescigno, A.A., Vaccaro, U. (eds) Combinatorial Algorithms. IWOCA 2024. Lecture Notes in Computer Science, vol 14764. Springer, Cham. https://doi.org/10.1007/978-3-031-63021-7_23

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