We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline g...

We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph G = (V,E) that can be solved as a linear minimum spanning tree problem. Characterization of such problems on graphs with special properties are given. This include complete graphs, complete bipartite graphs, cactuses among others. Our characterization can be verified in O(|E|) time. In the case of com...

Let h be a given positive integer. For a graph with n vertices and m edges, what is the maximum number of pairwise edge-disjoint induced subgraphs, each having minimum degree at least h? There are examples for which this number is O(m2/n2). We prove that this bound is achievable for all graphs with polynomially many edges. For all > 0, if m > n1+ , then there are always Ω(m2/n2) pairwise edge-d...

We propose a backtrack algorithm that solves a generalized version of the Maximum Induced Forest problem (MIF) in time O∗(1.8899n). The MIF problem is complementary to finding a minimum Feedback Vertex Set (FVS), a well-known intractable problem. Therefore the proposed algorithm can find a minimum FVS as well. To the best of our knowledge, this is the first algorithm that breaks the O∗(2n) barr...

In their seminal paper on geometric minimum spanning trees, Monma and Suri [11] showed how to embed any tree of maximum degree 5 as a minimum spanning tree in the Euclidean plane. The embeddings provided by their algorithm require area O(2n2) × O(2n2) and the authors conjectured that an improvement below cn×cn is not possible, for some constant c > 0. In this paper, we show how to construct MST...

Given a chromosome represented by a permutation of genes, a block-interchanges is proposed as a generalized transposition that affects a chromosome by swapping two nonintersecting segments of genes. The problem of sorting by block-interchanges is to find a minimum series of block-interchanges for transforming one chromosome into another. In this paper, we present an O(n+ δ log n) time algorithm...

We present a fast algorithm for uniform sampling of contiguous minimum cuts separating a source vertex from a set of sink vertices in a weighted undirected planar graph with n vertices embedded in the plane. The algorithm takes O(n) time per sample, after an initial O(n) preprocessing time during which the algorithm computes the number of all such contiguous minimum cuts. Contiguous cuts (that ...

Given a set P of n points in the plane, a Manhattan network of P is a network that contains a rectilinear shortest path between every pair of points of P . Aminimum Manhattan network of P is a Manhattan network of minimum total length. It is unknown whether it is NP-hard to construct a minimum Manhattan network. The best approximations published so far are a combinatorial 3-approximation algori...

Let (G) denote the minimum number of edges to be removed from a graph G to make it bipartite. For each 3-chromatic graph F we determine a parameter (F ) such that for each F-free graph G on n vertices with minimum degree (G) 2n/( (F )+ 2)+ o(n) we have (G)= o(n2), while there are F-free graphs H with (H) ≥ 2n/( (F )+ 2) for which (H)= (n2). © 2007 Elsevier B.V. All rights reserved. MSC: primary...

In this paper, we give a sufficient condition for a graph to contain vertex-disjoint stars of a given size. It is proved that if the minimum degree of the graph is at least k + t − 1 and the order is at least (t + 1)k + O(t), then the graph contains k vertex-disjoint copies of a star K1,t. The condition on the minimum degree is sharp, and there is an example showing that the term O(t) for the n...