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A new heuristic for finding verifiable k-vertex-critical subgraphs

Author

Listed:
  • Alex Gliesch

    (Federal University of Rio Grande do Sul)

  • Marcus Ritt

    (Federal University of Rio Grande do Sul)

Abstract

Given graph G, a k-vertex-critical subgraph (k-VCS) $$H \subseteq G$$ H ⊆ G is a subgraph with chromatic number $$\chi (H)=k$$ χ ( H ) = k , for which no vertex can be removed without decreasing its chromatic number. The main motivation for finding a k-VCS is to prove k is a lower bound on $$\chi (G)$$ χ ( G ) . A graph may have several k-VCSs, and the k-Vertex-Critical Subgraph Problem asks for one with the least possible vertices. We propose a new heuristic for this problem. Differently from typical approaches that modify candidate subgraphs on a vertex-by-vertex basis, it generates new subgraphs by a heuristic that optimizes for maximum edges. We show this strategy has several advantages, as it allows a greater focus on smaller subgraphs for which computing $$\chi $$ χ is less of a bottleneck. Experimentally the proposed method matches or improves previous results in nearly all cases, and more often finds solutions that are provenly k-VCSs. We find new best k-VCSs for several DIMACS instances, and further improve known lower bounds for the chromatic number in two open instances, also fixing their chromatic numbers by matching existing upper bounds.

Suggested Citation

  • Alex Gliesch & Marcus Ritt, 2022. "A new heuristic for finding verifiable k-vertex-critical subgraphs," Journal of Heuristics, Springer, vol. 28(1), pages 61-91, February.
  • Handle: RePEc:spr:joheur:v:28:y:2022:i:1:d:10.1007_s10732-021-09487-9
    DOI: 10.1007/s10732-021-09487-9
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    References listed on IDEAS

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