If G is a 2-connected bipartite biclaw-free graph with (G) 4, then by [2, Lemma 2.2], every edge of G lies in a 4-cycle, and then by Lemma 2.5 (the correct version), G is collapsible. It follows that G will have a spanning eulerian subgraph. Note that a Hamiltonian cycle of G is a spanning eulerian subgraph of G with maximum degree 2. We consider it one possible way to attack Conjecture 1.1 of ...

We present the rst truly polynomial-time approximation scheme (PTAS) for the minimum-cost k-vertex-(or, k-edge-) connected spanning subgraph problem for complete Euclidean graphs in R d : Previously it was known for every positive constant " how to construct in a polynomial time a graph on a superset of the input points which is k-vertex connected with respect to the input points, and whose cos...

We obtain a polynomial-time 17 12 -approximation algorithm for the minimum-cost 2-vertexconnected spanning subgraph problem, restricted to graphs of minimum degree at least 3. Our algorithm uses the framework of ear-decompositions for approximating connectivity problems, which was previously used in algorithms for finding the smallest 2-edge-connected spanning subgraph by Cheriyan, Sebo and Szi...

Answering a question of Adrian Bondy [4], we prove that every strong digraph has a spanning strong subgraph with at most n + 2α − 2 arcs, where α is the size of a maximum stable set of D. Such a spanning subgraph can be found in polynomial time. An infinite family of oriented graphs for which this bound is sharp was given by Odile Favaron [3]. A direct corollary of our result is that there exis...

We survey approximation algorithms and hardness results for versions of the Generalized Steiner Network (GSN) problem in which we seek to find a low cost subgraph (where the cost of a subgraph is the sum of the costs of its edges) that satisfies prescribed connectivity requirements. These problems include the following well known problems: min-cost k-flow, min-cost spanning tree, traveling sale...

By Petersen’s theorem, a bridgeless cubic multigraph has a 2-factor. H. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least 3 has a spanning even subgraph in which every component has at least four vertic...

• We give a 1.5-approximation for half-integral weighted MSCSS. Our matches known integrality gap lower bound. 2 − f approximation if LP values are bounded below by . In the minimum strongly connected spanning subgraph ( WMSCSS ) problem we must purchase minimum-cost of digraph. show that linear program (LP) solutions can be efficiently rounded to integral at multiplicative 1.5 cost. This round...

Given a complete graph with edge weights that satisfy the triangle inequality, we bound the minimum cost of a subgraph which is the union of two spanning trees in terms of the minimum cost of a k-edge-connected subgraph, for k ≤ 4.