Isomorph-free generation of 2-connected graphs with applications

Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely K_r-saturated graphs and find the list of uniquely K_4-saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to more problems concerning 2-connected graphs.Comment: 15 pages, 3 figures, 4 table

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Groebner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Groebner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.Comment: Discrete Math., to appea