A celebrated theorem of Savitch [Sav70] states that NSPACE(S) ⊆ DSPACE(S). In particular, Savitch gave a deterministic algorithm to solve ST-CONNECTIVITY (an NL-complete problem) using O(logn) space, implying NL ⊆ DSPACE(logn). While Savitch’s theorem itself has not been improved in the last four decades, studying the space complexity of several special cases of STCONNECTIVITY has provided new ...

In theoretical chemistry, -modified Wiener index is a graph invariant topological index to analyze the chemical properties of molecular structure. In this note, we determine the minimum -modified Wiener index of graph with fixed connectivity or edge-connectivity. Our results also present the sufficient and necessary condition for reaching the lower bound.

the stability of a communication network composed of processing nodes and communication links is of prime importance to network designers. as the network begins losing links or nodes, eventually there is a loss in its effectiveness. thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible...

The topic of network design occupies a central place in Combinatorial Optimization. There are a number of key open questions pertaining to k-node connected graphs on topics such as minimum-cost k-connected spanning subgraphs, connectivity augmentation, and orientations. These questions/conjectures are analogous to similar statements for edge connectivity. Most of the statements for edge connect...

We consider aspects of vertex splitting and connectivity augmentation in hypergraphs. In the splitting problem we are given a hypergraph G = (V + s, E) in which s is only incident with edges of size two and asked to “split” s in such a way that we preserve the local-connectivity in the set V . We develop results that lead to a characterisation of the hypergraphs in which there is no such split....

Graph connectivity theory is important in network implementations, transportation, routing and tolerance, among other things. Separation edges vertices refer to single points of failure a network, so they are often sought-after. Chandramouleeswaran et al. introduced the principle semiring valued graphs, also known as S-valued symmetry 2015. Since then, works on graphs such vertex dominating set...

Let G be a nontrivial connected graph of order n, and k an integer with 2 ≤ k ≤ n. For a set S of k vertices of G, let κ(S) denote the maximum number l of edge-disjoint trees T1, T2, . . . , Tl in G such that V (Ti) ∩ V (Tj) = S for every pair i, j of distinct integers with 1 ≤ i, j ≤ l. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by κk(G), o...

In Lecture 2, we saw how communication complexity lower bounds yield lower bounds for circuit depth. In particular, we showed that for any function f , D(KWf ) = depth(f) and D(KW f ) = depth (f), where KWf denotes the Karchmer-Wigderson game on f . Using this, we showed that monotone circuits for matching require Ω(n) depth. In this lecture we will show that circuits solving directed s-t conne...