Restricted domination in Quasi-transitive and 3-Quasi-transitive digraphs

Abstract

Let H be a digraph possibly with loops and D a digraph without loops with a coloring of its arcs c:A(D)→V(H) (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A subset N of vertices of D is said to be an H-kernel if (1) for every pair of different vertices in N there is no H-path between them and (2) for every vertex u in V(D)∖N there exists an H-path in D from u to N. Under this definition an H-kernel is a kernel whenever A(H)=0̸; an H-kernel is a kernel by monochromatic paths whenever A(H)={(u,u):u∈V(H)}; an H-kernel is a properly colored kernel whenever H has no loops and an H-kernel is a kernel by rainbow paths whenever H has no directed cycles.Let k be an integer, with k≥2. D is k-quasi-transitive if for every pair of vertices u and v of D, the existence of a directed path of length k from u to v implies that {(u,v),(v,u)}∩A(D)≠0̸.In Galeana-Sánchez and O’Reilly-Regueiro (2013), in Delgado-Escalante et al. (2018) and in Galeana-Sánchez and Rojas-Monroy (2006) the authors establish sufficient conditions to guarantee the existence of kernels by monochromatic paths in 3-quasi-transitive digraphs, the existence of properly colored kernels in 2-quasi-transitive digraphs and the existence of kernels in 2-quasi-transitive digraphs, respectively. In this paper we investigate the problem of the existence of H-kernels in quasi-transitive and in 3-quasi-transitive digraphs by means of a partition of V(H).We give some examples which show that each hypothesis in the main result for quasi-transitive digraphs is tight.