Abstract
Abstract In 2018, Bai, Fujita and Zhang [Discrete Math. 341 (2018), no. 6, 1523–1533] introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph D D , which is a subset S S of vertices of D D such that ( a a ) there exists no rainbow path for any pair of distinct vertices of S S , and ( b b ) every vertex outside S S can reach S S by a rainbow path in D D . They showed that it is NP-hard to recognize whether an arc-coloured digraph has an RP-kernel and it is NP-complete to decide whether an arc-coloured tournament has an RP-kernel. In this paper, we give the sufficient conditions for the existence of an RP-kernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments, and prove that these arc-coloured digraphs have RP-kernels if certain “short” cycles and certain “small” induced subdigraphs are rainbow.
Highlights
In 2018, Delgado-Escalante et al [14] gave some sufficient conditions for the existence of a kernel by properly coloured paths in arc-coloured tournaments, quasi-transitive digraphs and k-partite tournaments
Just as other NP-complete problems, we give some sufficient conditions for the existence of an RPkernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments and prove that these arc-coloured digraphs have RP-kernels if certain “short” cycles and certain “small” induced subdigraphs are rainbow
For an arc-coloured digraph D, the rainbow closure of D denoted by Cr(D), is a digraph such that: (a) V(Cr(D)) = V(D); (b) A(Cr(D)) = {(u, v) ∣ there exists a rainbow (u, v)-path in D}
Summary
In 2018, Delgado-Escalante et al [14] gave some sufficient conditions for the existence of a kernel by properly coloured paths in arc-coloured tournaments, quasi-transitive digraphs and k-partite tournaments. Define a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph D to be a subset S ⊆ V(D) such that (a) there exists no rainbow path for any pair of vertices of S, and (b) for each vertex outside S can reach S by a rainbow path. Just as other NP-complete problems, we give some sufficient conditions for the existence of an RPkernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments and prove that these arc-coloured digraphs have RP-kernels if certain “short” cycles and certain “small” induced subdigraphs are rainbow
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