Abstract
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.
Highlights
In this paper we study the computational complexity of DEFECTIVE COLORING, which is known in the literature as IMPROPER COLORING: given a graph and two parameters χd, ∆∗ we want to color the graph with χd colors so that every color class induces a graph with maximum degree at most ∆∗
Our focus on this paper is to study DEFECTIVE COLORING on subclasses of perfect graphs, which are perhaps the most widely studied class of graphs where GRAPH COLORING is in P
We present the following results: first, we show that DEFECTIVE COLORING is hard on split graphs even when ∆∗ is a fixed constant, as long as ∆∗ ≥ 1; the problem is in P if ∆∗ = 0
Summary
In this paper we study the computational complexity of DEFECTIVE COLORING, which is known in the literature as IMPROPER COLORING: given a graph and two parameters χd, ∆∗ we want to color the graph with χd colors so that every color class induces a graph with maximum degree at most ∆∗. For split graphs we show that DEFECTIVE COLORING is NP-hard, but that it remains NP-hard even if either χd or ∆∗ is a constant with the smallest possible non-trivial value (χd ≥ 2 or ∆∗ ≥ 1) To complement these negative results we provide a treewidth-based DP algorithm which runs in polynomial time if both χd and ∆∗ are constant, for split graphs, and for chordal graphs. This generalizes a previous algorithm of Havet et al (2009) on interval graphs. For the reader’s convenience, it depicts an inclusion diagram for the graph classes that we mention
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