Finding trails in multigraphs with restricted transitions

Abstract

Let G be a multigraph. A transition in G is a pair of adjacent edges. For each vertex x of G, a set T(x) of allowed transition with respect to x is a set of unordered pairs of edges incident to x. A transition system T is a set {T(x): x ∈ V(G)}, where T(x) is a fixed set of allowed transition with respect to x. Given a multigraph G and a transition system T, for each x ∈ V(G), the transition graph of x, denoted by G_x, is a graph such that its vertex set is the set of edges incident to x; and two vertices e and g of G_x are adjacent whenever eg ∈ T(x). A trail $P=(x_0,e_0,x_1,\dots ,x_{k-1},e_{k-1},x_k)$ in G is T-compatible if for every $i\in \{0,\dots ,k-2\}$, $e_i{e}_{i+1}\in T\left(x_{i+1}\right)$.

In this paper we deal with the problem of finding T-compatible trails between s and t two given vertices in a multigraph with transition system T. First, we prove that finding a T-compatible trail between two given edges can be done in polynomial time. Consequently, finding a T-compatible $s-t$ trail can be done in polynomial time. Moreover, it can be found a shortest T-compatible $s-t$ trail and a closed T-compatible trail containing a given vertex in polynomial time. Finally, we study multigraphs with transition systems such that their transition graph is connected or complete multipartite graph. The properly colored setting is a particular case of transition systems where all its transition graphs are complete multipartite graphs.