A graph is said to be (k, l)-connected if the resulted graph after removing of any k vertices and (l−1) edges or removing of any (k−1) vertices and l edges is still connected. Beineke and Harary (1967) (see [1]) claimed to prove that there should be k+ l edge-disjoint paths, of which k are vertex-disjoint, between any pair of vertices if the graph has (k, l)-connectivity. However, Mader (1979) ...

The Covering Steiner problem is a common generalization of the k-MST and Group Steiner problems. An instance of the Covering Steiner problem consists of an undirected graph with edge-costs, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the requ...

We study two-stage robust variants of combinatorial optimization problems like Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously studied by Dhamdhere et al. [9], Golovin et al. [15], and Feige et al. [13], are two-stage planning problems in which the requirements are revealed after some decisions are taken in stage one. One has to th...

A Steiner 2-design S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G on v vertices into Kks. The obvious necessary condition of those orders v for which there exists a halvable S(2,k,v) is that v admits the existence of an S(2,k,v) with an even number of blocks. In th...

Let G=(V; E) be a connected graph and S ⊂E. S is said to be a m-restricted edge cut (m-RC) if G − S is disconnected and each component contains at least m vertices. The m-restricted edge connectivity (G) is the minimum size of all m-RCs in G. Based on the fact that (G)6 3(G), where m(G)=min{!(X ): X ⊂V; |X |=m and G[X ] is connected} (!(X ) denotes the number of edges with one end vertex in X a...

Following Harary, the conditional connectivity (edge-connectivity) of a graph with respect to a given graph-theoretic property is the minimum cardinality of a set of vertices (edges), if any, whose deletion disconnects the graph and every remaining component has such a property. We study the case in which all these components are different from a tree whose order is not greater than n. For inst...

Let H $H$ be a k $k$ -uniform D $D$ -regular simple hypergraph on N $N$ vertices. Based an analysis of the Rödl nibble, in 1997, Alon, Kim and Spencer proved that if ⩾ 3 $k \geqslant 3$ , then contains matching covering all but at most − 1 / ( ) + o $ND^{-1/(k-1)+o(1)}$ vertices, asked whether this bound is tight. In paper we improve their by showing for > η $ND^{-1/(k-1)-\eta }$ vertices some ...

In this paper we propose and study a new complexity model for approximation algorithms. The main motivation are practical problems over large data sets that need to be solved many times for different scenarios, e.g., many multicast trees that need to be constructed for different groups of users. In our model we allow a preprocessing phase, when some information of the input graph G = (V,E) is s...

Each strongly minimal Steiner k-system (M, R) (where is R a ternary collinearity relation) can be ‘coordinatized’ in the sense of (Ganter–Werner 1975) by quasigroup if k prime-power. We show this coordinatization never definable and k-systems constructed (Baldwin–Paolini 2020) interpret quasigroup. Nevertheless, refining construction, prime power, each (2, k)-variety quasigroups (Definition 3.1...