Let be a graph. If there exists a spanning subgraph G F such that     1,3, , 2 1 F d x n    , then F is called to be -odd factor of . Some sufficient and necessary conditions are given for to have    n   1,2 1 G G U – 1,2 1 n  -odd factor where is any subset of such that U   V G U k  .

Suppose we are given a planar graph G with edge costs and we want to find a 2-edge-connected spanning subgraph of minimum cost. We present a polynomial time approximation scheme (PTAS) for this problem when the costs are uniform, or when the total cost of G is within a constant factor of the optimal.

A linear time algorithm for the Bottleneck Biconnected Spanning Subgraph problem is presented. This improves the hitherto best-known solution, which has a running time of 0( m + n log n), where m and n are the number of edges and vertices of the graph.

Halin proved in 1978 that there exists a normal spanning tree in every connected graph G that satisfies the following two conditions: (i) G contains no subdivision of a ‘fat’ Kא0 , one in which every edge has been replaced by uncountably many parallel edges; and (ii) G has no Kא0 subgraph. We show that the second condition is unnecessary.

The smallest number of edges that have to be deleted from a graph to obtain a bipartite spanning subgraph is called the bipartite edge frustration of G and denoted by φ(G). In this paper we determine the bipartite edge frustration of some classes of composite graphs. © 2010 Elsevier B.V. All rights reserved.

Let G = (V, E) be a graph with n vertices and m edges. The problem of constructing a spanning tree is to find a connected subgraph of G with n vertices and (n 1) edges. For a weighted graph, the minimum spanning tree problem can be solved in O(log m) time with O(m) processors on the CRCW PRAM, and for an unweighed graph, the spanning tree problem can be solved in O(log n) time with O(n +m) proc...

We consider the problem of deciding whether or not a geometric graph has a crossing-free spanning tree. This problem is known to be NP-hard even for very restricted types of geometric graphs. In this paper, we present an O(n) time algorithm to solve this problem for the special case of geometric graphs that arise as visibility graphs of a finite set of n points between two monotone polygonal ob...

A Halin graph is a simple plane graph consisting of a tree without degree 2 vertices and a cycle induced by the leaves of the tree. In 1975, Lovász and Plummer conjectured that every 4-connected plane triangulation has a spanning Halin subgraph. In this paper, we construct an infinite family of counterexamples to the conjecture.

We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k. The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation.

Let P be a set of n ≥ 3 points in general position in the plane and let G be a geometric graph with vertex set P . If the number of empty triangles 4uvw in P for which the subgraph of G induced by {u, v, w} is not connected is at most n− 3, then G contains a non-self intersecting spanning tree.