-kernels by walks in -colored digraphs and the color-class digraph. Issue 2 (1st August 2016) - Galeana-Sánchez, Hortensia; Sánchez-López, Rocío
Abstract: Let be a digraph possibly with loops and a finite digraph without loops whose arcs are colored with the vertices of ( is an -colored digraph). V( ) and A( ) will denote the sets of vertices and arcs of respectively. For an arc ( ) of we will denote by ( ) its color. A directed walk (respectively directed path) (, ) in is an -walk (respectively -path) if and only if ( ( ), ) is a directed walk in . A set is an -kernel by walks (respectively -kernel) if for every pair of different vertices in there is no -walk (respectively -path) between them, and for every vertex there exists such that there exists an -walk (respectively -path) from to in . Let be an arc-colored digraph. The color-class digraph of, denoted by ( ), is defined as follows: the vertices of the color-class digraph are the colors represented in the arcs of and ( ) A( ( )) if and only if there exist two arcs namely ( ) A( ) colored and ( ) A( ) colored . In this paper we relate the concepts discussed above, the color-class digraph and the -coloration of, in order to prove the existence of an -kernel by walks (respectively -kernel).