Defective Coloring on Classes of Perfect Graphs

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