Digraph Coloring and Distance to Acyclicity

In k-Digraph Coloring we are given a digraph and asked to partition its vertices into at most k sets, so that each set induces DAG. This well-known problem is NP-hard, as it generalizes (undirected) k-Coloring, but becomes trivial if the input acyclic. poses natural parameterized complexity question of what happens when “almost” this paper study using parameters measure input’s distance acyclicity in either directed or undirected sense. sense perhaps notion feedback vertex set. It already known that, for all ≥ 2, NP-hard on digraphs size + 4. We strengthen result show exactly k. immediately provides dichotomy, has − 1. Refining our reduction obtain three further consequences: (i) 2-Digraph oriented graphs 3; (ii) arc k2; interestingly, leads second FPT by k2 1; (iii) k, even maximum degree Δ 4k also almost tight, ≤ 3. Since these results imply bounded treewidth, then consider from underlying graph. On positive side, admits an algorithm whose parameter dependence (tw!)ktw. considerably worse than ktw pose whether tw! factor can be eliminated. Our main contribution part settle negative essentially optimal, much more restricted treedepth = 2. Specifically, solving with tdo(td) would contradict ETH. Then, class tournaments. deciding tournament 2-colorable NP-complete. present decides 2-color $O^{*}({\sqrt [3]{6}}^{n})$ time. Finally, explain how modified decide k-colorable.