A dichotomy for the kernel by $H$-walks problem in digraphs

TL;DR

This paper characterizes when the problem of finding kernels by H-walks in arc-colored digraphs is always solvable or NP-complete, based on properties of the digraph H.

Contribution

It establishes a dichotomy theorem classifying digraphs H as either panchromatic patterns or NP-complete cases for kernel existence.

Findings

Every digraph H is either a panchromatic pattern or NP-complete for kernel decision.

The paper provides a complete classification of the complexity for the kernel by H-walks problem.

The results unify and extend previous work on kernels in arc-colored digraphs.

Abstract

Let $H = (V_H, A_H)$ be a digraph which may contain loops, and let $D = (V_D, A_D)$ be a loopless digraph with a coloring of its arcs $c: A_D \to V_H$. An $H$-walk of $D$ is a walk $(v_0, \dots, v_n)$ of $D$ such that $(c(v_{i-1}, v_i), c(v_i, v_{i+1}))$ is an arc of $H$, for every $1 \le i \le n-1$. For $u, v \in V_D$, we say that $u$ reaches $v$ by $H$-walks if there exists an $H$-walk from $u$ to $v$ in $D$. A subset $S \subseteq V_D$ is a kernel by $H$-walks of $D$ if every vertex in $V_D \setminus S$ reaches by $H$-walks some vertex in $S$, and no vertex in $S$ can reach another vertex in $S$ by $H$-walks. A panchromatic pattern is a digraph $H$ such that every arc-colored digraph $D$ has a kernel by $H$-walks. In this work, we prove that every digraph $H$ is either a panchromatic pattern, or the problem of determining whether an arc-colored digraph $D$ has a kernel by $H$-walks is $NP$-complete.