Kernels by rainbow paths in arc-colored tournaments
Yandong Bai, Binlong Li, Shenggui Zhang

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 , define its {\em kernel by rainbow paths} to be a set of vertices such that (i) no two vertices of are connected by a rainbow path in , and (ii) every vertex outside can reach by a rainbow path in . 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 -vertex tournament with all its strongly connected -vertex subtournaments, , colored with at least colors has a kernel by rainbow paths, and the number of colors required cannot be reduced.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Complexity and Algorithms in Graphs
