We study the threshold for the existence of a spanning maximal planar subgraph in the random graph Gn . We show that it is very near p = ~TTOWe also discuss the threshold for the existence of a spanning maximal outerplanar subgraph. This is very near

A plane graph is a planar graph with a fixed embedding. Let G = (V,E) be an edge weighted connected plane graph, where V and E are the set of vertices and edges, respectively. Let F be the set of faces of G. For each edge e ∈ E, let w(e) ≥ 0 be the weight of the edge e of G. A face-spanning subgraph of G is a connected subgraph H induced by a set of edges S ⊆ E such that the vertex set of H con...

The main focus of this paper is the analysis of a simple randomized scheme for constructing low-weight k-connected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimum-weight k-connected spanning subgraph in a weighted complete graph, a NP-hard problem even if the weights satisfy the triangle inequality. We show that our algorithm gives...

Given a planar graph G the planar biconnectivity augmen tation problem is to add the minimum number of edges to G such that the resulting graph is still planar and biconnected Given a nonplanar and biconnected graph the maximum planar biconnected subgraph problem consists of removing the minimum number of edges so that planarity is achieved and biconnectivity is maintained Both problems are imp...

We introduce the minimum labelling spanning bi-connected subgraph problem (MLSBP) replacing connectivity by bi-connectivity in the well known minimum labelling spanning tree problem (MLSTP). A graph is bi-connected if, for every two vertices, there are, at least, two vertex-disjoint paths joining them. The problem consists in finding the spanning bi-connected subgraph or block with minimum set ...

For a graph H with at most n vertices and a weighing of the edges of Kn with integers, we seek a copy of H in Kn whose weight is minimal, possibly even zero. Of a particular interest are the cases where H is a spanning subgraph (or an almost spanning subgraph) and the case where H is a fixed graph. In particular, we show that relatively balanced weighings of Kn with {−r, . . . , r} guarantee al...

It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at lest five vertices, and the lower bound is best possible. A natural question arises whether we can improve the lower bound by changing the spanning property with the dominating property. In this paper, we show that a 3-edge-connected cubic graph has a dominating even subgraph in which eve...

We present the first polynomial-time approximation schemes, i.e., (1 + )-approximation algorithm for any constant > 0, for the minimum three-edge connected spanning subgraph problem and the minimum three-vertex connected spanning subgraph problem in undirected planar graphs. Both the approximation schemes run in linear time. This material is based upon work supported by the National Science Fou...

We consider the problem of locally constructing a spanning subgraph that approximates the Euclidean minimum spanning tree of a unit disk graph. We show that for any k ≥ 2 there exists a klocalized distributed algorithm that, given a unit disk graph U in the plane, constructs a planar subgraph of U containing a Euclidean MST on V (U), whose degree is at most 6, and whose total weight is at most ...