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Parameterized Mixed Graph Coloring

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  • Peter Damaschke

    (Chalmers University)

Abstract

Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and Vitaver) relates graph coloring to longest paths. It can be extended to mixed graphs. In the present paper we further extend the GHRV theorem to weighted mixed graphs. As a byproduct this yields a kernel and a parameterized algorithm (with the number of undirected edges as parameter) that is slightly faster than the brute-force algorithm. The parameter is natural since the directed version is polynomial whereas the undirected version is NP-complete. Furthermore we point out a new polynomial case where the edges form a clique.

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  • Peter Damaschke, 2019. "Parameterized Mixed Graph Coloring," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 362-374, August.
    • Handle: RePEc:spr:jcomop:v:38:y:2019:i:2:d:10.1007_s10878-019-00388-z
    DOI: 10.1007/s10878-019-00388-z
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    References listed on IDEAS

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    1. Bernard Gendron & Alain Hertz & Patrick St-Louis, 2007. "On edge orienting methods for graph coloring," Journal of Combinatorial Optimization, Springer, vol. 13(2), pages 163-178, February.
    2. Sotskov, Yu. N., 1991. "The complexity of shop-scheduling problems with two or three jobs," European Journal of Operational Research, Elsevier, vol. 53(3), pages 326-336, August.
    3. Pierre Hansen & Julio Kuplinsky & Dominique Werra, 1997. "Mixed graph colorings," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(1), pages 145-160, February.
    4. William W. Hardgrave & George L. Nemhauser, 1963. "A Geometric Model and a Graphical Algorithm for a Sequencing Problem," Operations Research, INFORMS, vol. 11(6), pages 889-900, December.
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    Cited by:

    1. Yuri N. Sotskov, 2020. "Mixed Graph Colorings: A Historical Review," Mathematics, MDPI, vol. 8(3), pages 1-24, March.

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