Small feedback vertex sets in planar digraphs

Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$, $|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This improves recent results of Golowich and Rolnick.Comment: 5 pages, 1 figure - v3 final versio

Digraph Coloring and Distance to Acyclicity

In $k$-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most $k$ sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) $k$-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question what happens when the input is "almost" acyclic. In this paper we study this question using parameters that measure the input's distance to acyclicity in either the directed or the undirected sense. It is already known that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard on digraphs of DFVS at most $k+4$. We strengthen this result to show that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for DFVS $k$. Refining our reduction we obtain two further consequences: (i) for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for graphs of feedback arc set (FAS) at most $k^2$; interestingly, this leads to a dichotomy, as we show that the problem is FPT by $k$ if FAS is at most $k^2-1$; (ii) $k$-Digraph Coloring is NP-hard for graphs of DFVS $k$, even if the maximum degree $\Delta$ is at most $4k-1$; we show that this is also almost tight, as the problem becomes FPT for DFVS $k$ and $\Delta\le 4k-3$. We then consider parameters that measure the distance from acyclicity of the underlying graph. We show that $k$-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is $(tw!)k^{tw}$. Then, we pose the question of whether the $tw!$ factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for $k=2$. Specifically, we show that an FPT algorithm solving $2$-Digraph Coloring with dependence $td^{o(td)}$ would contradict the ETH

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure