When k is even the min-max formula for the partition-constrained problem is a natural generalization of [3]. However this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge.

We prove that the Simplicity Preserving Edge-Connectivity Augmentation Problem and the problem of Increasing Edge-Connectivity by Reinforcing Edges are NP-complete. 1 Introduction The k-edge-connectivity augmentation problem is to find a smallest set F of new edges which makes a given (di)graph G = (V; E) k-edge-connected, that is, for which G 0 = (V; E [ F ) is k-edge-connected and jF j is as small as possible. (Note, that the augmented graph and also the augmenting set F itself may contain parallel edges.) This problem -- and a number of its extensions -- can be solved in polynomial time. More details on this and other connectivity augmentation problems can be found in the survey paper [3] by A. Frank. There are several interesting versions of this problem, however, which have not been solved yet by means of efficient algorithms. As it is noted in [3], "it is an important open problem to find algorithms which do not add parallel edges". This kind of simplicity requirement leads...

Splitting off two edges su, sv in a graph G means deleting su, sv and adding a new edge uv. Let G = (V + s, E) be k-edge-connected in V (k 2) and let d(s) be even. Lovasz proved that the edges incident to s can be split off in pairs in a such a way that the resulting graph on vertex set V is k-edge-connected. In this paper we investigate the existence of such complete splitting sequences when the set of split edges has to meet additional requirements. We prove structural properties of the set of those pairs u, v of neighbours of s for which splitting off su, sv destroys k-edge- connectivity. This leads to a new method for solving problems of this type.

Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies triangle inequalities, there are constant factor approximation algorithms for various degree bounded network design problems. • Global edge-connectivity: There is a (2 + 1 k)approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. • Local edge-connectivity: There is a 6-approximation algorithm for the minimum bounded degree Steiner network problem. • Global vertex-connectivity: There is a (2 + k−1 n + 1 k)-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem. • Spanning tree: There is an (1 + 1 d−1)-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with smallest possible maximum degree, and the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides’ algorithm for metric TSP. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to “short-cut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

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Splitting off a pair su; sv of edges in a graph G means the operation that deletes su and sv and adds a new edge uv. Given a graph G = (V + s; E) which is k-edge-connected (k 2) between vertices of V and a specified subset R ` V , first we consider the problem of finding a longest possible sequence of disjoint pairs (splittings) of edges sx; sy, x; y 2 R which can be split off preserving k-edge-connectivity in V . If R = V and d(s) is even then the well-known splitting off theorem of Lovasz asserts that a complete R-splitting exists, that is, all the edges connecting s to R can be split off in pairs. This is not the case in general. We characterize the graphs possessing a complete R-splitting and give a formula for the length of a longest R-splitting sequence. The main result of our paper is a solution for the following optimization problem: given G and R as above, find a smallest set F of new edges incident to s such that G 0 = (V + s; E + F ) has a complete R-splitting....

Let A(n; k; t) denote the smallest integer e for which every k-connected graph on n vertices can be made (k + t)-connected by adding e new edges. We determine A(n; k; t) for all values of n; k and t in the case of (directed and undirected) edge-connectivity and also for directed vertex-connectivity. For undirected vertexconnectivity we determine A(n; k; 1) for all values of n and k. We also describe the graphs which attain the extremal values. 1 Introduction B. Bollob'as posed the following problem in his book Extremal Graph Theory in 1978, see [3, Page 49, Problem 34]. Let 1 k ! l ! n. Determine the minimal integer e for which to every k-connected graph of order n it is possible to add at most e edges such that the resulting graph is l-connected. Determine the analogous minimum for edge-connectivity. In this paper we determine this number e for all values of n; k; l in the case of (directed and undirected) edge-connectivity and directed vertex-connectivity. In the undirected ver...

Splitting off a pair of edges su; sv in a graph G means replacing these two edges by a new edge uv. This operation is well-known in graph theory. Let G = (V + s; E + F ) be a graph which is k-edge-connected in V and suppose that jF j is even. Here F denotes the set of edges incident with s. Lovasz [12] proved that if k 2 then the edges in F can be split off in pairs preserving the k-edge-connectivity in V . This result was recently extended to the case where a bipartition R [ Q = V is given and every split edge must connect R and Q [4]. In this paper we investigate an even more general problem, where two disjoint subsets R; Q ae V are given and the goal is to split off (a largest possible subset of) the edges of F preserving k-edge-connectivity in V in such a way that every split edge incident with a vertex from R has the other end-vertex in Q. Motivated by connectivity augmentation problems, we introduce another extension, the so-called split completion version of our prob...

We consider the 2-edge-connectivity augmentation problem: given a graph S = (V, E) which is not 2-edge-connected and a set of new edges E ′ ⊆ V × V with non-negative weights, find a minimum cost subset X of E ′ such that adding the edges of X to S results in a 2-edge-connected graph. A practical application is the extension of an existing telecommunication network to become robust against single link failures. We compare, experimentally, different algorithms for solving general and large-scale instances. This includes exact methods based on mathematical programming, simple construction heuristics and metaheuristics. As part of the design of heuristics, we consider different neighborhood structures for local search, among which a very large scale neighborhood. In all cases, we exploit approaches through the graph formulation as well as through an equivalent set covering formulation. The results indicate that exact solutions by means of a basic integer programming model can be obtained in reasonably short time even on networks with 800 vertices and around 287.000 edges. Alternatively, an advanced heuristic algorithm based on subgradient optimization and iterated greedy finds often the optimal solution and is very fast. All previous benchmark instances are easily solved to optimality and new, larger, instances are introduced and studied.

Abstract. In this paper we present new edge splitting-off results maintaining all-pairs edge-connectivities of a graph. We first give an alternate proof of Mader’s theorem, and use it to obtain a deterministic Õ(rmax 2 · n 2)-time complete edge splitting-off algorithm for unweighted graphs, where rmax denotes the maximum edge-connectivity requirement. This improves upon the best known algorithm by Gabow by a factor of ˜Ω(n). We then prove a new structural property, and use it to further speedup the algorithm to obtain a randomized Õ(m + rmax 3 · n)-time algorithm. These edge splitting-off algorithms can be used directly to speedup various graph algorithms. 1