6391408
Computing the Scattering Number of Graphs
The scattering number of a noncomplete connected graph G is defined by s ( G )= max{ y ( G m X ) m X X ² V ( G ), y ( G m X ) S 2}, where y ( G m X ) denotes the number of components of G m X . This parameter can be used to measure the vulnerability of networks. It shows not only the difficulty to break down the network but also the damage that has been caused. In this paper, we prove that the problem of computing the scattering number of a graph is NP-complete. At the same time, the scattering numbers of grids and those of Cartesian products of two complete graphs are determined.
6437552
Heavy paths and cycles in weighted graphs
A weighted graph is a graph in which each edge is assigned a non-negative number, called the weight. The weight of a path (cycle) is the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with the vertex. A usual (unweighted) graph can be considered as a weighted graph with constant weight 1. In this paper, it is proved that for a 2-connected weighted graph, if every vertex has weighted degree at least d, then for any given vertex y, either y is contained in a cycle with weight at least 2d or every heaviest cycle is a Hamilton cycle. This result is a common generalization of Grotschel's theorem and Bondy–Fan's theorem assuring the existence of a cycle with weight at least 2d on the same condition. Also, as a tool for proving this result, we show a result concerning heavy paths joining two specific vertices and passing through one given vertex.
1154359
Scattering number in graphs
The scattering number of a noncomplete connected graph G is defined by s(G) = max{ω(G − X) − |X|: X ⊂ V(G), ω(G − X) ≥ 2}, where ω(G − X) denotes the number of components of the graph G − X. In this paper, we show that this parameter can be used to measure the vulnerability of a graph. To some extent, it represents a trade-off between the amount of work done to damage the network and how badly the network is damaged. The relationship between the scattering number and some other parameters of a graph is discussed. Furthermore, we give the Nordhaus–Gaddum-type result for scattering number. © 2001 John Wiley & Sons, Inc.
25468199
Edge vulnerability parameters of split graphs
Abstract A graph is called a split graph if its vertex set can be partitioned into a clique and an independent set. In this work, we investigate three vulnerability parameters of split graphs when edges are removed, i.e., edge-connectivity, edge-toughness and edge-integrity. It is proved that, for a noncomplete connected split graph G , its edge-connectivity is δ ( G ) , and its edge-toughness is min { δ ( G ) , | E ( G ) | | V ( G ) | − 1 } , where δ ( G ) , V ( G ) and E ( G ) , are the minimum degree, the vertex set and the edge set of G , respectively. Furthermore, we show that the edge-integrity of a noncomplete connected split graph equals its order when its minimum degree is greater than half of the size of its largest clique.
39027321
A Fan Type Condition For Heavy Cycles in Weighted Graphs
Abstract. A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree dw(v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. max{dw(x),dw(y)∣d(x,y)=2}≥c/2; 2. w(xz)=w(yz) for every vertex z∈N(x)∩N(y) with d(x,y)=2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least c. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result.
13607587
Heavy cycles passing through some specified vertices in weighted graphs
A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2-connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2-connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35
3196759
Low-complexity algorithms for sequencing jobs with a fixed number of job-classes
In this paper we consider the problem of scheduling n jobs such that makespan is minimized. It is assumed that the jobs can be divided into K job-classes and that the change-over time between two consecutive jobs depends on the job-classes to which the two jobs belong. In this setting, we discuss the one machine scheduling problem with arbitrary processing times and the parallel machines scheduling problem with identical processing times. In both cases it is assumed that the number of job-classes K is fixed. By using an appropriate integer programming formulation with a fixed number of variables and constraints, it is shown that these two problems are solvable in polynomial time. For the one machine scheduling case it is shown that the complexity of our algorithm is linear in the number of jobs n. Moreover, if the problem is encoded according to the high multiplicity model of Hochbaum and Shamir, the time complexity of the algorithm is shown to be a polynomial in log n. In the parallel machine scheduling case, it is shown that if the number of machines is fixed the same results hold.
15381284
Color Rapid Prototyping for Diffusion-Tensor MRI Visualization
We describe work toward creating color rapid prototyping (RP) plaster models as visualization tools to support scientific research in diffusion-tensor (DT) MRI analysis. We currently give surgeons and neurologists virtual-reality (VR) applications to visualize different aspects of their brain data, but having physical representations of those virtual models allows them to review the data with a very robust, natural, and fast haptic interface: their own hands. Our initial results are encouraging, and end users are excited about the possibilities of this technique. For example, using these models in conjunction with digital models on the computer screen or VR environment provides a static frame of reference that helps keep users oriented during their analysis tasks.
5300524
FastR: fast database search tool for non-coding RNA
The discovery of novel non-coding RNAs has been among the most exciting recent developments in biology. Yet, many more remain undiscovered. It has been hypothesized that there is in fact an abundance of functional non-coding RNA (ncRNA) with various catalytic and regulatory junctions. Computational methods tailored specifically for ncRNA are being actively developed. As the inherent signal for ncRNA is weaker than that for protein coding genes, comparative methods offer the most promising approach, and are the subject of our research. We consider the following problem: Given an RNA sequence with a known secondary structure, efficiently compute all structural homologs (computed as a function of sequence and structural similarity) in a genomic database. Our approach, based on structural filters that eliminate a large portion of the database, while retaining the true homologs allows us to search a typical bacterial database in minutes on a standard PC, with high sensitivity and specificity. This is two orders of magnitude better than current available software for the problem.
14478113
Sequencing jobs that require common resources on a single machine: A solvable case of the TSP
In this paper a one-machine scheduling model is analyzed wheren different jobs are classified intoK groups depending on which additional resource they require. The change-over time from one job to another consists of the removal time or of the set-up time of the two jobs. It is sequence-dependent in the sense that the change-over time is determined by whether or not the two jobs belong to the same group. The objective is to minimize the makespan. This problem can be modeled as an asymmetric Traveling Salesman Problem (TSP) with a specially structured distance matrix. For this problem we give a polynomial time solution algorithm that runs in O(n logn) time. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
31017610
Structural Alignment of Pseudoknotted RNA
In this paper, we address the problem of discovering novel non-coding RNA (ncRNA) using primary sequence, and secondary structure conservation, focusing on ncRNA families with pseudo-knotted structures. Our main technical result is an efficient algorithm for computing an optimum structural alignment of an RNA sequence against a genomic substring. This algorithm finds two applications. First, by scanning a genome, we can identify novel (homologous) pseudoknotted ncRNA, and second, we can infer the secondary structure of the target aligned sequence. We test an implementation of our algorithm (Pal), and show that it has near-perfect behavior for predicting the structure of many known pseudoknots. Additionally, it can detect the true homologs with high sensitivity and specificity in controlled tests. We also use Pal to search entire viral genome and mouse genome for novel homologs of some viral, and eukaryotic pseudoknots respectively. In each case, we have found strong support for novel homologs.