Augmenting a (k - 1)-Vertex-Connected Multigraph ℓ-Edge-Connected and k-Vertex-Connected Multigraph

Optimal Augmentation of a 2-Vertex-Connected Multigraph to a k-Edge-Connected and 3-Vertex-Connected Multigraph

Generating k-Vertex Connected Spanning Subgraphs and k-Edge Connected Spanning Subgraphs

We show that k-vertex connected spanning subgraphs of a given graph can be generated in incremental polynomial time for any fixed k. We also show that generating k-edge connected spanning subgraphs, where k is part of the input, can be done in incremental polynomial time. These results are based on properties of minimally k-connected graphs which might be of independent interest.

Generating Minimal k-Vertex Connected Spanning Subgraphs

We show that minimal k-vertex connected spanning subgraphs of a given graph can be generated in incremental polynomial time for any fixed k.

Minimally (k, k-1)-edge-connected graphs

For an interger l > 1, the l-edge-connectivity λl(G) of G is defined to be the smallest number of edges whose removal leaves a graph with at least l components, if |V (G)| ≥ l; and λl(G) = |V (G)| if |V (G)| ≤ l. A graph G is (k, l)-edge-connected if the l-edge-connectivity of G is at least k. A sufficient and necessary condition for G to be minimally (k, k − 1)-edgeconnected is obtained in the...

Vertex Covers and Connected Vertex Covers in 3-connected Graphs

A vertex cover of a graph G=(V,E) is a subset N of V such that each element of E is incident upon some element of N, where V and E are the sets of vertices and of edges of G, respectively. A connected vertex cover of a graph G is a vertex cover of G such that the subgraph G[N] induced by N of G is connected. The minimum vertex cover problem (VCP) is the problem of finding a vertex cover of mini...