K-connected Set Research Articles

Characterising k-connected sets in infinite graphs

A k-connected set in an infinite graph, where k>0 is an integer, is a set of vertices such that any two of its subsets of the same size ℓ≤k can be connected by ℓ disjoint paths in the whole graph.We characterise the existence of k-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. We also prove a duality theorem for the existence of such k-connected sets: if a graph contains no such k-connected set, then it has a tree-decomposition which, whenever it exists, precludes the existence of such a k-connected set.

Improved approximation algorithms for k-connected m-dominating set problems

A graph is k-connected if it has k pairwise internally node disjoint paths between every pair of its nodes. A subset S of nodes in a graph G is a k-connected set if the subgraph G[S] induced by S is k-connected; S is an m-dominating set if every v∈V∖S has at least m neighbors in S. If S is both k-connected and m-dominating then S is a k-connectedm-dominating set, or (k,m)-cds for short. In the k-Connectedm-Dominating Set ((k,m)-CDS) problem the goal is to find a minimum weight (k,m)-cds in a node-weighted graph. We consider the case m≥k and obtain the following approximation ratios. For unit disc graphs we obtain ratio O(kln⁡k), improving the ratio O(k2ln⁡k) of [1,2]. For general graphs we obtain the first non-trivial approximation ratio O(k2ln⁡n).