A SAT Approach for Finding Sup-Transition-Minors

The cycle double cover conjecture is a famous longstanding unsolved conjecture in graph theory. It is related and can be reduced to the compatible circuit decomposition problem. Recently Fleischner et al. (2018) provided a sufficient condition for a compatible circuit decomposition, which is called SUD-\(K_5\)-minor freeness. In a previous work we developed an abstract mathematical model for finding SUD-\(K_5\)-minors and based on the model a mixed integer linear program (MIP). In this work we propose a respective boolean satisfiability (SAT) model and compare it with the MIP model in computational tests. Non-trivial symmetry breaking constraints are proposed, which improve the solving times of both models considerably. Compared to the MIP model the SAT approach performs significantly better. We use the faster algorithm to further test graphs of graph theoretic interest and were able to get new insights. Among other results we found snarks with 30 and 32 vertices that do not contain a perfect pseudo-matching, that is a spanning subgraph consisting of \(K_2\) and \(K_{1,3}\) components, whose contraction leads to a SUD-\(K_5\)-minor free graph.

[1]  Mikolás Janota,et al.  On the Quest for an Acyclic Graph , 2017, RCRA@AI*IA.

[2]  Igor L. Markov,et al.  Solving difficult SAT instances in the presence of symmetry , 2002, Proceedings 2002 Design Automation Conference (IEEE Cat. No.02CH37324).

[3]  Cun-Quan Zhang,et al.  Circuit Decompositions of Eulerian Graphs , 2000, J. Comb. Theory, Ser. B.

[4]  C. Godsil,et al.  Cycles in graphs , 1985 .

[5]  G. Szekeres,et al.  Polyhedral decompositions of cubic graphs , 1973, Bulletin of the Australian Mathematical Society.

[6]  Brendan D. McKay,et al.  Practical graph isomorphism, II , 2013, J. Symb. Comput..

[7]  Melody Chan.  A survey of the cycle double cover conjecture , 2009 .

[8]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[9]  Günther R. Raidl,et al.  A model for finding transition-minors , 2020, Discret. Appl. Math..

[10]  C. Sims.  Computational methods in the study of permutation groups , 1970 .

[11]  Herbert Fleischner,et al.  Eulersche linien und Kreisüberdeckungen, die vorgegebene durchgänge in den Kanten vermeiden , 1980, J. Comb. Theory, Ser. B.

[12]  Marc E. Pfetsch,et al.  Branch-Cut-and-Propagate for the Maximum k-Colorable Subgraph Problem with Symmetry , 2011, CPAIOR.

[13]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[14]  Cun-Quan Zhang,et al.  Cycle covers (III) - Compatible circuit decomposition and K5-transition minor , 2019, J. Comb. Theory, Ser. B.