Given a set V of elements, S a family of subsets of V , and G a connected graph on vertex set S,a connected set cover (CSC) is a subfamily R of S such that every element in V is covered by at least one set of R, and the subgraph G[R] of G induced by R is connected. If furthermore G[R] is k-connected and every element in V is covered by at least m sets inR, thenR is a (k,m)-CSC. In this paper, w...

A flaw in the greedy approximation algorithm proposed by Zhang et al. for minimum connected set cover problem is corrected, and a stronger result on the approximation ratio of the modified greedy algorithm is established. The results are now consistent with the existing results on connected dominating set problem which is a special case of the minimum connected set cover problem.

In this paper, a certain new connectedness of L-fuzzy subsets inL-topological spaces is introduced and studied by means of preclosed sets. Itpreserves some fundamental properties of connected set in general topology.Especially the famous K. Fan’s Theorem holds.

The topic is the average order A(G) of a connected induced subgraph graph G. This generalizes, to graphs in general, subtree tree. In 1983 Jamison proved that order, over all trees n, minimized by path Pn, being A(Pn)=(n+2)/3. 2018, Kroeker, Mol, and Oellermann conjectured Pn minimizes G - conjecture was recently proved. this short note we show lower bound can be improved if connectivity known....