Cut covers of acyclic digraphs

TL;DR

This paper investigates the conditions under which acyclic digraphs can be covered by a certain number of cuts, establishing optimal degree bounds, extremal examples, and the computational complexity of related decision problems.

Contribution

It proves the optimality of degree bounds for covering acyclic digraphs with cuts, identifies extremal examples, and shows NP-completeness of the covering decision problem for certain digraphs.

Findings

Degree condition for covering acyclic digraphs by k cuts is optimal.

Powers of directed paths serve as extremal examples for certain parameters.

Deciding if a digraph admits a k-cut cover is NP-complete, even for planar graphs with bounded degrees.

Abstract

A cut in a digraph $D=(V,A)$ is a set of arcs $\{uv \in A: u\in U, v\notin U\}$, for some $U\subseteq V$. It is known that the arc set $A$ is covered by $k$ cuts if and only if it admits a $k$-coloring such that no two consecutive arcs $uv, vw$ receive the same color. Alon, Bollob\'as, Gy\'arf\'as, Lehel and Scott (2007) observed that every acyclic digraph of maximum indegree at most $\binom{k}{\lfloor k/2 \rfloor}-1$ is covered by $k$ cuts. We prove that this degree condition is best possible (if an enormous outdegree is allowed). Notably, for $k\geq 5$, powers of directed paths do not suffice as extremal examples. Instead, we locate the maximum $d$ such that the $d$-th power of an arbitrarily long directed path is covered by $k$ cuts between $(1-o(1)) \frac{1}{e} 2^k$ and $\frac{1}{2}2^k-2$. Let $k\geq 3$ and $D$ be an acyclic digraph that is not covered by $k$ cuts. We prove that the decision problem whether a digraph that admits a homomorphism to $D$ is covered by $k$ cuts is NP-complete. If $k=3$ and $D$ is the third power of the directed path on 12 vertices, then even the restriction to planar digraphs of maximum indegree and outdegree $3$ holds.