Enumerating Eulerian Trails via Hamiltonian Path Enumeration

Hiroyuki Hanada, Shuhei Denzumi, Yuma Inoue, Hiroshi Aoki, Norihito Yasuda, Shogo Takeuchi, Shin-ichi Minato

Abstract: Given an undirected graph G, we consider enumerating all Eulerian trails, that is, walks containing each of the edges in G just once. We consider achieving it with the enumeration of Hamiltonian paths with the zero-suppressed decision diagram (ZDD), a data structure that can efficiently store a family of sets satisfying given conditions. First we compute the line graph L(G), the graph representing adjacency of the edges in G. We also formulated the condition when a Hamiltonian path in L(G) corresponds to an Eulerian trail in G because every trail in G corresponds to a path in L(G) but the converse is not true. Then we enumerate all Hamiltonian paths in L(G) satisfying the condition with ZDD by representing them as their sets of edges.