Abstract
For an arc-colored digraph D, define its 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 first show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a 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⩽k⩽n, colored with at least k−1 colors has a kernel by rainbow paths.
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