On Generation of Cut Conjunctions, Minimal K-connected Spanning Subgraphs, Minimal Connected and Spanning Subsets and Vertices

OF THE DISSERTATION On Generation of Cut Conjunctions, Minimal k-Connected Spanning Subgraphs, Minimal Connected and Spanning Subsets and Vertices by Konrad Borys Dissertation Director: Professor Endre Boros We consider the following problems: • Cut conjunctions in graphs: given an undirected graphG = (V,E) and a collection of vertex pairs B ⊆ V × V generate all minimal edge sets X ⊆ E such that every pair (s, t) ∈ B of vertices is disconnected in (V,E rX), which is a special case of the more general cut conjunctions problem in matroids: given a matroid M on ground set S = E ∪B, generate all minimal subsets X ⊆ E such that no element b ∈ B is spanned by E rX. We give an incremental polynomial time algorithm to generate cut conjunctions in graphs. In contrast, the generation of cut conjunctions for vectorial matroids turns out to be NP-hard. • k-Vertex (k-edge) connected spanning subgraphs: given an undirected graph G = (V,E), generate all minimal edge sets X ⊆ E such that the graph (V,X) is k-vertex (k-edge) connected. We show that k-vertex connected spanning subgraphs can be generated in incremental polynomial time for any fixed k. The running time of our algorithm is is