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.
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References
Wilson, R.J.: Introduction to Graph Theory, 4th edn. Pearson Education (1996)
Harary, F.: Graph Theory, 1st edn. Addison-Wesley (1969)
Mihail, M., Winkler, P.: On the number of Eulerian orientations of a graph. In: Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1992, pp. 138–145 (1992)
Creed, P.: Sampling Eulerian orientations of triangular lattice graphs. Journal of Discrete Algorithms 7(2), 168–180 (2009)
Ge, Q., Štefankovič, D.: The complexity of counting Eulerian tours in 4-regular graphs. Algorithmica 63(3), 588–601 (2012)
Kasteleyn, P.W.: A soluble self-avoiding walk problem. Physica 29(12), 1329–1337 (1963)
Rubin, F.: A search procedure for Hamilton paths and circuits. Journal of the ACM 21(4), 576–580 (1974)
Mateti, P., Deo, N.: On algorithms for enumerating all circuits of a graph. SIAM Journal on Computing 5(1), 90–99 (1976)
van der Zijpp, N.J., Catalano, S.F.: Path enumeration by finding the constrained k-shortest paths. Transportation Research Part B: Methodological 39(6), 545–563 (2005)
Liu, H., Wang, J.: A new way to enumerate cycles in graph. In: International Conference on Internet and Web Applications and Services/Advanced International Conference on Telecommunications, p. 57 (2006)
Minato, S.-i.: Zero-suppressed BDDs and their applications. International Journal on Software Tools for Technology Transfer 3(2), 156–170 (2001)
Diestel, R.: Graph Theory, 4th edn. Springer (2010)
Chartrand, G.: On Hamiltonian line-graphs. Transactions of the American Mathematical Society 134, 559–566 (1968)
Harary, F., Nash-Williams, C.S.J.A.: On Eulerian and Hamiltonian graphs and line graphs. Canadian Mathematical Bulletin 8, 701–709 (1965)
Knuth, D.E.: 7.1.4 Binary Decision Diagrams. In: Combinatorial Algorithms, vol. 4A. The Art of Computer Programming, vol. 4A. Pearson Education (2011)
Knuth, D.E.: Don Knuth’s home page, http://www-cs-staff.stanford.edu/~uno/
Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers C-35(8), 677–691 (1986)
Inoue, T., Iwashita, H., Kawahara, J., Minato, S.: Graphillion: Software library designed for very large sets of graphs in python. Technical Report TCS-TR-A-13-65, Division of Computer Science, Hokkaido University (2013)
Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating-sign matrices and domino tilings (part I). Journal of Algebraic Combinatorics 1(2), 111–132 (1992)
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Hanada, H. et al. (2015). Enumerating Eulerian Trails via Hamiltonian Path Enumeration. In: Rahman, M.S., Tomita, E. (eds) WALCOM: Algorithms and Computation. WALCOM 2015. Lecture Notes in Computer Science, vol 8973. Springer, Cham. https://doi.org/10.1007/978-3-319-15612-5_15
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DOI: https://doi.org/10.1007/978-3-319-15612-5_15
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