Trails in arc-colored digraphs avoiding forbidden transitions

Let H be a digraph possibly with loops. Let D be a digraph without loops. An H -coloring of D is a function c : A ( D ) → V ( H ) . We say that D is an H -colored digraph whenever we are taking a fixed H -coloring of D . A trail W = ( v 0 , e 0 , v 1 , e 1 , v 2 , . . . , v n − 1 , e n − 1 , v n ) in D is an H -trail if and only if ( c ( e i ) , c ( e i +1 )) is an arc in H for every i ∈ { 0 , . . . , n − 2 } . Whenever the vertices of an H -trail are pairwise different, we say that it is an H -path. In this paper, we study the problem of finding s − t H -trail in H -colored digraphs. First, we prove that finding an H -trail starting with the arc e and ending at arc f can be done in polynomial time. As a consequence, we give a polynomial time algorithm to find the shortest H -trail from a vertex s to a vertex t (if it exists). Moreover, we obtain a Menger-type theorem for H -trails. As a consequence, we show that the problem of maximizing the number of arc disjoint s − t H -trails in D can be solved in polynomial time. Although finding an H -path between two given vertices is an NP-problem, it becomes a polynomial time problem in the case when H is a reflexive and transitive digraph.