## Rémy Belmonte ; Michael Lampis ; Valia Mitsou - Defective Coloring on Classes of Perfect Graphs

dmtcs:4926 - Discrete Mathematics & Theoretical Computer Science, January 20, 2022, vol. 24, no. 1 - https://doi.org/10.46298/dmtcs.4926
Defective Coloring on Classes of Perfect Graphs

Authors: Rémy Belmonte ; Michael Lampis ; Valia Mitsou

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.

Volume: vol. 24, no. 1
Section: Discrete Algorithms
Published on: January 20, 2022
Accepted on: December 12, 2021
Submitted on: October 29, 2018
Keywords: Computer Science - Data Structures and Algorithms,Mathematics - Combinatorics