We show that for pairs (Q,R) and (S,T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V(G)−(Q∪R∪S∪T) such that there are two ways to reduce G by a vertex-minor operation while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids. Joint work with Sang-il Oum.
On August 10, 2021, Duksang Lee (이덕상) from KAIST and IBS Discrete Mathematics Group gave a talk at the Discrete Math Seminar, showing that there are finitely many pivot-minor-minimal graphs preserving the rank connectivity between two fixed pairs of vertex sets. The title of his talk was “Intertwining connectivities for vertex-minors and pivot-minors“.
On November 24, 2020, Duksang Lee (이덕상) from KAIST and the IBS Discrete Mathematics Group gave a talk on the characterization of matroids whose bases form graphic delta-matroids at the Discrete Math Seminar. The title of his talk was “Characterizing matroids whose bases form graphic delta-matroids“.
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most 48 elements. This is joint work with Sang-il Oum.