Kernels by properly colored paths in arc-colored digraphs

Abstract

A kernel by properly colored paths of an arc-colored digraph D is a set S of vertices of D such that (i) no two vertices of S are connected by a properly colored directed path in D, and (ii) every vertex outside S can reach S by a properly colored directed path in D. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.

Kernels by rainbow paths in arc-colored tournaments

“…In general, things look better in dense instances. For example, we shall show that the PCP-kernel problem in tournaments is polynomially solvable, but in general digraphs it has been proved to be NP-hard in [1]. However, for the rainbow kernel problem, it will be shown to be NP-complete even when we restrict the digraph class to arc-colored tournaments.…”

“…Chvátal [7] showed in 1973 that it is NP-complete to decide whether a digraph has a kernel. Bai et al [1] pointed out in 2018 that it is NP-hard to recognize whether an arc-colored digraph has a rainbow kernel. In this paper we study the complexity of the rainbow kernel problem in arc-colored tournaments.…”

“…In the concluding section of [1], Bai et al introduced the notion of rainbow kernels but did not present detailed analysis. Here a kernel by rainbow paths (or rainbow kernel for short) of an arc-colored digraph D is defined, similar to the definition of MP-kernels or PCP-kernels, 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 concentrate on rainbow kernels in arc-colored tournaments.…”

“…A 2-colored tournament T * 5 with no rainbow kernel.In[1] Bai et al provided a sufficient condition for the existence of a PCP-kernel in arccolored tournaments. Another main contribution of this paper concentrates on the complexity of the above problem.…”

H-kernels in unions of H-colored quasi-transitive digraphs

“…an H-kernel is a kernel by monochromatic paths (mp-kernel) (a subset N of vertices of D such that (1) for every u and v in N there exists no monochromatic directed paths between u and v and (2) for every u in V (D) \ N there exists v in N such that there exists a monochromatic directed path from u to v) and when H has no loops, an H-kernel is an alternating kernel (a subset N of vertices of D such that (1) for every u and v in N it holds that there exists no directed path between u and v in which consecutive arcs have different colors and (2) for every u in V (D) \ N there exists v in N such that there is a directed path from u to v in which consecutive arcs have different colors). In each of these special cases for H, sufficient conditions have been established in order to guarantee the existence of H-kernels, see for example [3,5,7,9,15]. Thus we can see that the concept of H-kernels is a generalization of the concepts of kernels, mp-kernels and alternating kernels.…”