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.
1204298
An implicit degree condition for long cycles in 2-connected graphs
Abstract Let id ( v ) denote the implicit degree of a vertex v . In this work we prove that: If G is a 2-connected graph with max { id ( u ) , id ( v ) } ≥ c / 2 for each pair of nonadjacent vertices u and v that are vertices of an induced claw or an induced modified claw of G , then G contains either a Hamilton cycle or a cycle of length at least c . This extends several previous results on the existence of long cycles in graphs.
6040880
Rupture degree of graphs
We introduce a new graph parameter, the rupture degree. The rupture degree for a complete graph K n is defined as 1−n, and the rupture degree for an incomplete connected graph G is defined by r(G)=max{ω(G−X)−|X|−m(G−X):X⊂V(G), ω(G−X)>1}, where ω(G−X) is the number of components of G−X and m(G−X) is the order of a largest component of G−X. It is shown that this parameter can be used to measure the vulnerability of networks. Rupture degrees of several specific classes of graphs are determined. Formulas for the rupture degree of join graphs and some bounds of the rupture degree are given. We also obtain some Nordhaus–Gaddum type results for the rupture degree.
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.
40273501
Pattern Discovery in Probabilistic Databases
Modeling probabilistic data is one of important issues in databases due to the fact that data is often uncertainty in real-world applications. So, it is necessary to identify potentially useful patterns in probabilistic databases. Because probabilistic data in 1NF relations is redundant, previous mining techniques don't work well on probabilistic databases. For this reason, this paper proposes a new model for mining probabilistic databases. A partition is thus developed for preprocessing probabilistic data in a probabilistic databases. We evaluated the proposed technique, and the experimental results demonstrate that our approach is effective and efficient.
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.
11213421
Consensus Folding of Unaligned RNA Sequences Revisited
As one of the earliest problems in computational biology, RNA secondary structure prediction (sometimes referred to as "RNA folding") problem has attracted attention again, thanks to the recent discoveries of many novel non-coding RNA molecules. The two common approaches to this problem are de novo prediction of RNA secondary structure based on energy minimization and the consensus folding approach (computing the common secondary structure for a set of unaligned RNA sequences). Consensus folding algorithms work well when the correct seed alignment is part of the input to the problem. However, seed alignment itself is a challenging problem for diverged RNA families. In this paper, we propose a novel framework to predict the common secondary structure for unaligned RNA sequences. By matching putative stacks in RNA sequences, we make use of both primary sequence information and thermodynamic stability for prediction at the same time. We show that our method can predict the correct common RNA secondary structures even when we are given only a limited number of unaligned RNA sequences, and it outperforms current algorithms in sensitivity and accuracy.
16330147
A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming
This paper presents a new and high performance solution method for multistage stochastic convex programming. Stochastic programming is a quantitative tool developed in the field of optimization to cope with the problem of decision-making under uncertainty. Among others, stochastic programming has found many applications in finance, such as asset-liability and bond-portfolio management. However, many stochastic programming applications still remain computationally intractable because of their overwhelming dimensionality. In this paper we propose a new decomposition algorithm for multistage stochastic programming with a convex objective and stochastic recourse matrices, based on the path-following interior point method combined with the homogeneous self-dual embedding technique. Our preliminary numerical experiments show that this approach is very promising in many ways for solving generic multistage stochastic programming, including its superiority in terms of numerical efficiency, as well as the flexibility in testing and analyzing the model.
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.