A dichotomy for the kernel by H‐walks problem in 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 → V H. An H‐walk of D is a walk ( v 0 , … , 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 ≤ i ≤ n − 1. For u , v ∈ 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 ⊆ V D is a kernel by H‐walks of D if every vertex in V D \ 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 H‐arc‐colored digraph D has a kernel by H‐walks. In this study, 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 N P‐complete.

Panchromatic patterns by paths

“…Proof. If F 1 is a panchromatic pattern by paths, the proof of this result is found in [7]. Otherwise, the proof is analogous (cf.…”

“…We now turn our attention to B3 . In [7], a characterization of panchromatic patterns (by walks) was given in terms of forbidden subdigraphs and, although it is based on the characterization of B 3 found in [8] (which we mentioned earlier is flawed), a similar approach can be used to describe the structure of the panchromatic patterns by paths. The idea is to classify all the patterns on three vertices; knowing which of them are panchromatic patterns by paths gives us enough information to describe the general structure of patterns in this family.…”

“…Nonetheless, the authors of the present work recently found that the proof of a crucial lemma in the aforementioned work is incorrect, and thus, the characterization of this class remains as an open problem. Galeana-Snchez and Hernández-Cruz used this flawed characterization in [7] to prove that every pattern H is either a panchromatic pattern, or the problem of determining whether an H-arc-coloured digraph has a kernel by H-walks is N P -complete; luckily, the main argument is robust enough to remain true even if the family of panchromatic patterns by H-walks turns out to be a family different from the one proposed in [8].…”

$H$-kernels in $H$-colored digraphs without $(ξ_{1}, ξ, ξ_{2})$-$H$-subdivisions of $\overrightarrow{C_{3}}$

“…In [4] we find that it is NP-complete to recognize whether a digraph has a kernel. In [7] the authors prove that the problem of determining whether an H-colored digraph has a kernel by H-walks is in NP. In [3] we find that (1) it is NP-hard to recognize whether an arc-colored digraph has a PCP-kernel and (2) it is NP-hard to recognize whether an arc-colored digraph has a kernel by rainbow paths.…”

Minimal digraph obstructions for small matrices

“…Although it is already known that the M -partition problem for such patterns is polynomial time solvable, it is useful to have the exact list of minimal M -obstructions. Such list has been already used in [6] (where it was meant to be originally included), and recently in [4].…”