Kernels by rainbow paths in arc-colored tournaments

TL;DR

This paper investigates the computational complexity and conditions for the existence of kernels by rainbow paths in arc-colored tournaments, revealing NP-completeness and specific coloring conditions that guarantee such kernels.

Contribution

It proves NP-completeness for deciding the existence of kernels by rainbow paths in arc-colored tournaments and establishes coloring conditions that ensure their existence.

Findings

Deciding kernel existence is NP-complete.

Certain coloring conditions guarantee kernels by rainbow paths.

Number of colors needed cannot be reduced below a threshold.

Abstract

For an arc-colored digraph $D$, define its {\em kernel by rainbow paths} to be a set $S$ of vertices such that (i) no two vertices of $S$ are connected by a rainbow path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a rainbow path in $D$. In this paper, we show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a {\em tournament} is an orientation of a complete graph. In addition, we show that every arc-colored $n$-vertex tournament with all its strongly connected $k$-vertex subtournaments, $3\leq k\leq n$, colored with at least $k-1$ colors has a kernel by rainbow paths, and the number of colors required cannot be reduced.