Unavoidable Parallel Minors of 4-Connected Graphs

A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K_{4,k} with a complete graph on the vertices of degree k, the k-partition triple fan with a complete graph on the vertices of degree k, the k-spoke double wheel, the k-spoke double wheel with axle, the (2k+1)-rung Mobius zigzag ladder, the (2k)-rung zigzag ladder, or K_k. We also find the unavoidable parallel minors of 1-, 2-, and 3-connected graphs.Comment: 12 pages, 3 figure

Forbidden Directed Minors and Kelly-width

Partial 1-trees are undirected graphs of treewidth at most one. Similarly, partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known that an undirected graph is a partial 1-tree if and only if it has no K_3 minor. In this paper, we generalize this characterization to partial 1-DAGs. We show that partial 1-DAGs are characterized by three forbidden directed minors, K_3, N_4 and M_5

Bipartite Minors

We introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain $K_{3,3}$ as a bipartite minor. Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests. We then establish a recursive characterization of bipartite $(2,2)$-Laman graphs --- a certain family of graphs that contains all maximal bipartite planar graphs.Comment: 9 page