Enumerating Eulerian Trails via Hamiltonian Path Enumeration

• Published in: WALCOM: Algorithms and Computation pp. 161 - 174
• Main Authors: Hanada, Hiroyuki, Denzumi, Shuhei, Inoue, Yuma, Aoki, Hiroshi, Yasuda, Norihito, Takeuchi, Shogo, Minato, Shin-ichi
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Eulerian trail; Hamiltonian path; line graph; path enumeration; zero-suppressed binary decision diagram
• ISBN: 9783319156118; 331915611X
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-15612-5_15

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 graphL(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.