Eulerian Paths with Regular Constraints

Labeled graphs, in which edges are labeled by letters from some alphabet Σ, are extensively used to model many types of relations associated with actions, costs, owners, or other properties. Each path in a labeled graph induces a word in Σ∗ – the one obtained by concatenating the letters along the edges in the path. Classical graph-theory problems give rise to new problems that take these words into account. We introduce and study the constrained Eulerian path problem. The input to the problem is a Σ-labeled graph G and a specification L ⊆ Σ∗. The goal is to find an Eulerian path in G that satisfies L. We consider several classes of the problem, defined by the classes of G and L. We focus on the case L is regular and show that while the problem is in general NP-complete, even for very simple graphs and specifications, there are classes that can be solved efficiently. Our results extend work on Eulerian paths with edge-order constraints. We also study the constrained Chinese postman problem, where edges have costs and the goal is to find a cheapest path that contains each edge at least once and satisfies the specification. Finally, we define and study the Eulerian language of a graph, namely the set of words along its Eulerian paths. 1998 ACM Subject Classification G.2.2 Graph Theory