We consider the problems of finding the minimum-weight 2-connected spanning subgraph in edge-weighted planar graphs and its variations. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs. From this we derive quasi-polynomial time ap...

In 1991, Bollobás and Frieze showed that the threshold for Gn,p to contain a spanning maximal planar subgraph is very close to p = n−1/3. In this paper, we compute similar threshold ranges for Gn,p to contain a maximal bipartite planar subgraph and for Gn,p to contain a maximal planar subgraph of fixed girth g.

Consider the minimum k-edge-connected spanning subgraph problem: given a positive integer k and a k-edge-connected graph G, nd a k-edge-connected spanning subgraph of G with minimum number of edges. This problem is known to be NP-complete. Khuller and Raghavachari presented the rst algorithm with a performance ratio smaller than 2 for all k. They proved an upper bound of 1.85 for the performanc...

G. R. Grimmett and S. N. Winkler Abstra t. We consider three probability measures on subsets of edges of a given finite graph G, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connec...

In the (k, λ)-subgraph problem, we are given an undirected graph G = (V, E) with edge costs and two positive integers k and λ and the goal is to find a minimum cost simple λ-edge-connected subgraph of G with at least k nodes. This generalizes several classical problems, such as the minimum cost k-Spanning Tree problem or k-MST (which is a (k, 1)-subgraph), and minimum cost λ-edge-connected span...

Given a graph with positive edge weights and a positive integer m, the Constrained Forest Problem (CFP) problem seeks a minimum-weight spanning subgraph each of whose components contains at least m vertices. Such a subgraph is called an m-forest. We present a genetic algorithm (GA) for CFP, which is NP-hard for me”4. Our GA evolves good spanning trees, as determined by the weight of the m-fores...

Given a polyhedron P which is of interest, a major goal of polyhedral combinatorics is to find classes of essential, i.e. facet inducing inequalities which describe P . In general this is a difficult task. We consider the case in which we have knowledge of facets for a face F of P , and present some general theory and methods for exploiting the close relationship between such polyhedra in order...

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph G consisting of (1 + ǫ)n edges (for a prespecified constant ǫ > 0...

This note concerns the f -parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function f : V (G) → N, and our goal is to find a spanning subgraph F of G with degF ≤ f and minimizing the number of vertices x with degF (x) ≡ f(x) mod 2. First we prove a Gallai–Edmonds type structure theorem and some other known results on the f -parity subgraph problem, ...