Independent restricted domination and the line digraph

Abstract

Let H be a digraph possibly with loops and let D be a digraph whose arcs are colored with the vertices of H (an H-colored digraph). A walk (path) P in D will be called an H-restricted walk (path) if the colors displayed on the arcs of P form a walk in H. An H-restricted kernel N is a set of vertices of D such that for every two different vertices in N there is no H-restricted path in D joining them, and for every x in V(D) — N there exists an H-restricted path in D from x to N.For the line digraph of D we consider its inner arc-coloration, defined as follows: If h is an arc of D with color c then any arc of the form (x, h) in L(D) also has color c.We prove that the number of H-restricted kernels in an H-colored digraph is equal to the number of H-restricted kernels in the inner coloration of its line digraph.