Bounds and Fixed-Parameter Algorithms for Weighted Improper Coloring

Guðmundsson, Bjarki Ágúst; Magnússon, Tómas Ken

doi:10.1016/j.entcs.2016.03.013

Cited by 5 publications

(2 citation statements)

References 22 publications

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“…This has led to the study of the problem on Unit Disk Graphs Havet et al (2009) as well as various classes of grids Araújo et al (2012); Bermond et al (2010); Archetti et al (2015). Weighted generalizations have also been considered Bang-Jensen and Halldórsson (2015); Gudmundsson et al (2016). More background can be found in the survey by Frick (1993) or Kang's PhD thesis (Kang (2008)).…”

Section: Introductionmentioning

confidence: 99%

Defective Coloring on Classes of Perfect Graphs

Belmonte¹,

Lampis²,

Mitsou³

2022

Discrete Mathematics &Amp; Theoretical Computer Science

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In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded.

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Section: Introductionmentioning

confidence: 99%

Defective Coloring on Classes of Perfect Graphs

Belmonte¹,

Lampis²,

Mitsou³

2022

Discrete Mathematics &Amp; Theoretical Computer Science

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show abstract

“…This leads to the concept of relaxed (or improper, or defective) coloring of a graph. Please refer to [1,3,5].…”

Section: Introductionmentioning

confidence: 99%

On t-relaxed coloring of complete multi-partite graphs

Lan,

Lin

2021

Preprint

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Let G be a graph and t a nonnegative integer. Suppose f is a mapping from the vertex set of G to {1, 2, . . . , k}. If, for any vertex u of G, the number of neighbors v of u with f (v) = f (u) is less than or equal to t, then f is called a t-relaxed k-coloring of G. And G is said to be (k, t)-colorable. The t-relaxed chromatic number of G, denote by χ t (G), is defined as the minimum integer k such that G is (k, t)-colorable. A set S of vertices in G is t-sparse if S induces a graph with a maximum degree of at most t. Thus G is (k, t)-colorable if and only if the vertex set of G can be partitioned into k t-sparse sets. It was proved by Belmonte, Lampis and Mitsou (2017) that the problem of deciding if a complete multi-partite graph is (k, t)-colorable is NP-complete. In this paper, we first give tight lower and up bounds for the t-relaxed chromatic number of complete multi-partite graphs. And then we design an algorithm to compute maximum t-sparse sets of complete multi-partite graphs running in O((t + 1) 2 ) time. Applying this algorithm, we show that the greedy algorithm for χ t (G) is 2-approximate and runs in O(tn) time steps (where n is the vertex number of G). In particular, we prove that for t ∈ {1, 2, 3, 4, 5, 6}, the greedy algorithm produces an optimal t-relaxed coloring of a complete multi-partite graph. While, for t ≥ 7, examples are given to illustrate that the greedy strategy does not always construct an optimal t-relaxed coloring.

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Defective Coloring on Classes of Perfect Graphs

Belmonte

Lampis

Mitsou

2017

Graph-Theoretic Concepts in Computer Science

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Abstract. In Defective Coloring we are given a graph G and two integers χ d , ∆ * and are asked if we can χ d -color G so that the maximum degree induced by any color class is at most ∆ * . We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters χ d , ∆ * is set to the smallest possible fixed value that does not trivialize the problem (χ d = 2 or ∆ * = 1). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either χ d or ∆ * is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both χ d and ∆ * are unbounded.

show abstract

Bounds and Fixed-Parameter Algorithms for Weighted Improper Coloring

Cited by 5 publications

References 22 publications

Defective Coloring on Classes of Perfect Graphs

Defective Coloring on Classes of Perfect Graphs

On t-relaxed coloring of complete multi-partite graphs

Defective Coloring on Classes of Perfect Graphs

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