-kernels by walks in -colored digraphs and the color-class digraph
Hortensia Galeana‐Sánchez, Rocío Sánchez-López · AKCE International Journal of Graphs and Combinatorics · 2016
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).