The computational complexity of weighted vertex coloring for \(\{P_5,K_{2,3},K^+_{2,3}\}\)-free graphs

Abstract

In this paper, we show that the weighted vertex coloring problem can be solved in polynomial on the sum of vertex weights time for \(\{P_5,K_{2,3}, K^+_{2,3}\}\)-free graphs. As a corollary, this fact implies polynomial-time solvability of the unweighted vertex coloring problem for \(\{P_5,K_{2,3},K^+_{2,3}\}\)-free graphs. As usual, \(P_5\) and \(K_{2,3}\) stands, respectively, for the simple path on 5 vertices and for the biclique with the parts of 2 and 3 vertices, \(K^+_{2,3}\) denotes the graph, obtained from a \(K_{2,3}\) by joining its degree 3 vertices with an edge.

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Acknowledgements

Research is supported under financial support of Russian Science Foundation, project No 19-71-00005.

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Correspondence to D. S. Malyshev.

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Malyshev, D.S., Razvenskaya, O.O. & Pardalos, P.M. The computational complexity of weighted vertex coloring for \(\{P_5,K_{2,3},K^+_{2,3}\}\)-free graphs. Optim Lett 15, 137–152 (2021). https://doi.org/10.1007/s11590-020-01593-0

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Keywords

  • Coloring problem
  • Hereditary class
  • Computational complexity