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Acyclic Coloring Parameterized by Directed Clique-Width

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Algorithms and Discrete Applied Mathematics (CALDAM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12601))

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Abstract

An acyclic r-coloring of a directed graph \(G=(V,E)\) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the dichromatic number equals the well-known chromatic number of the underlying undirected graph. This allows us to carry over the \(\text {W}[1]\)-hardness and lower bounds for running times of the chromatic number problem parameterized by clique-width to the dichromatic number problem parameterized by directed clique-width. We introduce the first polynomial-time algorithm for the acyclic coloring problem on digraphs of constant directed clique-width. From a parameterized point of view our algorithm shows that the Dichromatic Number problem is in \(\text {XP}\) when parameterized by directed clique-width and extends the only known structural parameterization by directed modular width for this problem. Furthermore, we apply defineability within monadic second order logic in order to show that Dichromatic Number problem is in \(\text {FPT}\) when parameterized by the directed clique-width and r. For directed co-graphs, which is a class of digraphs of directed clique-width 2, we even show a linear time solution for computing the dichromatic number.

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Notes

  1. 1.

    XP is the class of all parameterized problems which can be solved by algorithms that are polynomial if the parameter is considered as a constant [9].

  2. 2.

    FPT is the class of all parameterized problems which can be solved by algorithms that are exponential only in the size of a fixed parameter while being polynomial in the size of the input size [9].

  3. 3.

    The proofs of the results marked with a \(\bigstar \) are omitted due to space restrictions, see [20].

  4. 4.

    We use the notion of a multi set, i.e., a set that may have several equal elements. For a multi set with elements \(x_1,\ldots ,x_n\) we write \(\mathcal{M}=\langle x_1,\ldots ,x_n \rangle \). The number how often an element x occurs in \(\mathcal{M}\) is denoted by \(\psi (\mathcal{M},x)\). Two multi sets \(\mathcal{M}_1\) and \(\mathcal{M}_2\) are equal if for each element \(x \in \mathcal{M}_1 \cup \mathcal{M}_2\), \(\psi (\mathcal{M}_1,x)=\psi (\mathcal{M}_2,x)\), otherwise they are called different. The empty multi set is denoted by \(\langle \rangle \).

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Acknowledgments

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 388221852.

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Authors and Affiliations

  1. Institute of Computer Science, Algorithmics for Hard Problems Group, Heinrich-Heine-University Düsseldorf, 40225, Düsseldorf, Germany

    Frank Gurski, Dominique Komander & Carolin Rehs

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  1. Frank Gurski
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Correspondence to Frank Gurski .

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Editors and Affiliations

  1. Indian Institute of Technology Ropar, Rupnagar, India

    Apurva Mudgal

  2. Institute of Mathematical Sciences, Chennai, India

    C. R. Subramanian

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Gurski, F., Komander, D., Rehs, C. (2021). Acyclic Coloring Parameterized by Directed Clique-Width. In: Mudgal, A., Subramanian, C.R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2021. Lecture Notes in Computer Science(), vol 12601. Springer, Cham. https://doi.org/10.1007/978-3-030-67899-9_8

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  • DOI: https://doi.org/10.1007/978-3-030-67899-9_8

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