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Parameterized mixed graph coloring

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Title: Parameterized mixed graph coloring
Authors: Damaschke, Peter, 1963
Source: Journal of Combinatorial Optimization. 38(2):362-374
Subject Terms: longest path, parameterized algorithm, graph coloring, scheduling, mixed graph
Description: 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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  Data: Parameterized mixed graph coloring
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  Data: <searchLink fieldCode="AR" term="%22Damaschke%2C+Peter%22">Damaschke, Peter</searchLink>, 1963
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  Data: <i>Journal of Combinatorial Optimization</i>. 38(2):362-374
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  Data: <searchLink fieldCode="DE" term="%22longest+path%22">longest path</searchLink><br /><searchLink fieldCode="DE" term="%22parameterized+algorithm%22">parameterized algorithm</searchLink><br /><searchLink fieldCode="DE" term="%22graph+coloring%22">graph coloring</searchLink><br /><searchLink fieldCode="DE" term="%22scheduling%22">scheduling</searchLink><br /><searchLink fieldCode="DE" term="%22mixed+graph%22">mixed graph</searchLink>
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  Data: 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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        Value: 10.1007/s10878-019-00388-z
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    Subjects:
      – SubjectFull: longest path
        Type: general
      – SubjectFull: parameterized algorithm
        Type: general
      – SubjectFull: graph coloring
        Type: general
      – SubjectFull: scheduling
        Type: general
      – SubjectFull: mixed graph
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      – TitleFull: Parameterized mixed graph coloring
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              Type: published
              Y: 2019
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