Certain generalization of the maximum traveling salesman problem

The problem is considered of finding in a complete undirected weighted graph a connected spanning subgraph with the given degrees of the vertices having maximum total weight of the edges. An approximate polynomial algorithm is presented for this problem. The algorithm is analyzed, and some upper bounds of its error are established in the general case as well as in the metric and Euclidean cases.     

一类三圈图的Wiener指数

图G的Wiener指数是指图G中所有顶点对间的距离之和,即W（G）=∑dc（u,u）,{u,u}CG其中de（u,u）表示G中顶点u,u之间的距离.三圈图是指边数与顶点数之差等于2的连通图,任意两个圈至多只有一个公共点的三圈图记为T_n~3.研究了三圈图T_n~3的Wiener指数,给出了其具有最小、次小Wiener指数的图结构.     

On vertex symmetric digraphs

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω(epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph (e?p?n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K_m in K_p and equicolored graphs in K_p.     

Class Steiner trees and VLSI-design

We consider a generalized version of the Steiner problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with required classes of vertices and Steiner vertices, find a shortest connected subgraph containing at least one vertex of each required class. We show that this problem is NP-hard, even if there are no Steiner vertices and the graph is a tree. Moreover, the same complexity result holds if the input class Steiner graph additionally is embedded in a unit grid, if each vertex has degree at most three, and each class consists of no more than three vertices. For similar restricted versions, we prove MAX SNP-hardness and we show that there exists no polynomial-time approximation algorithm with a constant bound on the relative error, unless P = NP. We propose two efficient heuristics computing different approximate solutions in time O(¦E¦+¦V¦log¦V¦) and in time O(c(¦E¦+¦V¦log¦V¦)), respectively, where E is the set of edges in the given graph, V is the set of vertices, and c is the number of classes. We present some promising implementation results. kw]Steiner Tree; Heuristic; Approximation complexity; MAX-SNP-hardness     

Minimal comparability completions of arbitrary graphs

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes Π for which Π completion of arbitrary graphs can be achieved through such a vertex incremental approach.     

The product representation of a locally dependent random graph

Directed graphs with random black and white colourings of edges such that the colours of edges from different vertices are mutually independent are called locally dependent random graphs. Two random graphs are equivalent if they cannot be distinguished from percolation processes on them if only the vertices are seen. A necessary and sufficient condition is given for when a locally dependent random graph is equivalent to a product random graph; that is one in which the edges can be grouped in such a way that within each group the colours of the edges are equivalent and between groups they are independent. As an application the random graph corresponding to a spatial general epidemic model is considered.     

图的最小完整度

主要研究了图的完整度,并给出若干关于完整度的结果. 对于所有的顶点数和边数都给定的连通图类,如何确定该图类中完整度最小的图. 同时研究了对于顶点数和完整度都给定的连通图类,如何确定该图类中边数最多的图的问题. 这些结果为图的最小完整度的优化设计提供了理论和方法.     

Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs

Multicut is a fundamental network communication and connectivity problem. It is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NP-completeness and (fixed-parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth.     

Register Allocation in Structured Programs

In this article we look at the register allocation problem. In the literature this problem is frequently reduced to the general graph coloring problem and the solutions to the problem are obtained from graph coloring heuristics. Hence, no algorithm with a good performance guarantee is known. Here we show that when attention is restricted tostructured programswhich we define to be programs whose control-flow graphs are series-parallel, there is an efficient algorithm that produces a solution which is within a factor of 2 of the optimal solution. We note that even with the previous restriction the resulting coloring problem is NP-complete.We also consider how to delete a minimum number of edges from arbitrary control-flow graphs to make them series-parallel and to apply our algorithm. We show that this problem is Max SNP hard. However, we define the notion of anapproximate articulation pointand we give efficient algorithms to find approximate articulation points. We present a heuristic for the edge deletion problem based on this notion which seems to work well when the given graph is close to series-parallel.     

The backup 2-median problem on block graphs

The backup 2-median problem is a location problem to locate two facilities at vertices with the minimum expected cost where each facility may fail with a given probability. Once a facility fails, the other one takes full responsibility for the services. Here we assume that the facilities do not fail simultaneously. In this paper, we consider the backup 2-median problem on block graphs where any two edges in one block have the same length and the lengths of edges on different blocks may be different. By constructing a tree-shaped skeleton of a block graph, we devise an O（n log n q- m）-time algorithm to solve this problem where n and m are the number of vertices and edges, respectively, in the given block graph.     

Minimal split completions

We study the problem of adding an inclusion minimal set of edges to a given arbitrary graph so that the resulting graph is a split graph, called a minimal split completion of the input graph. Minimal completions of arbitrary graphs into chordal and interval graphs have been studied previously, and new results have been added recently. We extend these previous results to split graphs by giving a linear-time algorithm for computing minimal split completions. We also give two characterizations of minimal split completions, which lead to a linear time algorithm for extracting a minimal split completion from any given split completion.We prove new properties of split graph that are both useful for our algorithms and interesting on their own. First, we present a new way of partitioning the vertices of a split graph uniquely into three subsets. Second, we prove that split graphs have the following property: given two split graphs on the same vertex set where one is a subgraph of the other, there is a sequence of edges that can be removed from the larger to obtain the smaller such that after each edge removal the modified graph is split.     

三圈图的Harary指数

图G的Harary指数是指图G中所有顶点对间的距离倒数之和. 三圈图是指边数等于顶点数加2的连通图. 研究了三圈图的Harary数, 给出了所有三圈图中具有极大Harary指数的图的结构以及含有三个圈的三圈图中具有次大Harary指数的图的结构.     

An Efficient Branch-and-bound Algorithm for Finding a Maximum Clique with Computational Experiments

We present an exact and efficient branch-and-bound algorithm MCR for finding a maximum clique in an arbitrary graph. The algorithm is not specialized for any particular type of graph. It employs approximate coloring to obtain an upper bound on the size of a maximum clique along with an improved appropriate sorting of vertices. We demonstrate by computational experiments on random graphs with up to 15,000 vertices and on DIMACS benchmark graphs that in general, our algorithm decidedly outperforms other existing algorithms. The algorithm has been successfully applied to interesting problems in bioinformatics, image processing, design of quantum circuits, and design of DNA and RNA sequences for biomolecular computation.     

A separator theorem for graphs of bounded genus

Many divide-and-conquer algorithms on graphs are based on finding a small set of vertices or edges whose removal divides the graph roughly in half. Most graphs do not have the necessary small separators, but some useful classes do. One such class is planar graphs: If an n-vertex graph can be drawn on the plane, then it can be bisected by removal of O(sqrt(n)) vertices (R. J. Lipton and R. E. Tarjan, SIAM J. Appl. Math.36 (1979), 177–189). The main result of the paper is that if a graph can be drawn on a surface of genus g, then it can be bisected by removal of O(sqrt(gn)) vertices. This bound is best possible to within a constant factor. An algorithm is given for finding the separator that takes time linear in the number of edges in the graph, given an embedding of the graph in its genus surface. Some extensions and applications of these results are discussed.     

Subexponential parameterized algorithms for degree-constrained subgraph problems on planar graphs

We present subexponential parameterized algorithms on planar graphs for a family of problems of the following shape: given a graph, find a connected (induced) subgraph with bounded maximum degree and with maximum number of edges (or vertices). These problems are natural generalisations of the Longest Path problem. Our approach uses bidimensionality theory combined with novel dynamic programming techniques over branch decompositions of the input graph. These techniques can be applied to a more general family of problems that deal with finding connected subgraphs under certain degree constraints.     

A solitaire game played on 2-colored graphs

We introduce a solitaire game played on a graph. Initially one disk is placed at each vertex, one face green and the other red, oriented with either color facing up. Each move of the game consists of selecting a vertex whose disk shows green, flipping over the disks at neighboring vertices, and deleting the selected vertex. The game is won if all vertices are eliminated. We derive a simple parity-based necessary condition for winnability of a given game instance. By studying graph operations that construct new graphs from old ones, we obtain broad classes of graphs where this condition also suffices, thus characterizing the winnable games on such graphs. Concerning two familiar (but narrow) classes of graphs, we show that for trees a game is winnable if and only if the number of green vertices is odd, and for n-cubes a game is winnable if and only if the number of green vertices is even and not all vertices have the same color. We provide a linear-time algorithm for deciding winnability for games on maximal outerplanar graphs. We reduce the decision problem for winnability of a game on an arbitrary graph G to winnability of games on its blocks, and to winnability on homeomorphic images of G obtained by contracting edges at 2-valent vertices.     

An Ant Colony Optimization Algorithm for the Minimum Weight Vertex Cover Problem

Given an undirected graph and a weighting function defined on the vertex set, the minimum weight vertex cover problem is to find a vertex subset whose total weight is minimum subject to the premise that the selected vertices cover all edges in the graph. In this paper, we introduce a meta-heuristic based upon the Ant Colony Optimization (ACO) approach, to find approximate solutions to the minimum weight vertex cover problem. In the literature, the ACO approach has been successfully applied to several well-known combinatorial optimization problems whose solutions might be in the form of paths on the associated graphs. A solution to the minimum weight vertex cover problem however needs not to constitute a path. The ACO algorithm proposed in this paper incorporates several new features so as to select vertices out of the vertex set whereas the total weight can be minimized as much as possible. Computational experiments are designed and conducted to study the performance of our proposed approach. Numerical results evince that the ACO algorithm demonstrates significant effectiveness and robustness in solving the minimum weight vertex cover problem.     

Infinite quantum graphs

Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.     

Geometric Graphs with No Three Disjoint Edges

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position and edges are represented by straight line segments. We show that a geometric graph on n vertices with no three pairwise disjoint edges has at most 2.5n edges. This result is tight up to an additive constant.     

Error graphs and the reconstruction of elements in groups

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an evolutionary process. It can be defined as the reconstruction, or identification, of an unknown vertex in a given graph from a minimal number of vertices (erroneous or distorted patterns) in a metric ball of a given radius r around the unknown vertex. For this problem it is required to find minimum restrictions for such a reconstruction to be possible and also to find efficient reconstruction algorithms under such minimal restrictions.In this paper we define error graphs and investigate their basic properties. A particular class of error graphs occurs when the vertices of the graph are the elements of a group, and when the path metric is determined by a suitable set of group elements. These are the undirected Cayley graphs. Of particular interest is the transposition Cayley graph on the symmetric group which occurs in connection with the analysis of transpositional mutations in molecular biology [P.A. Pevzner, Computational Molecular Biology: An Algorithmic Approach, MIT Press, Cambridge, MA, 2000; D. Sankoff, N. El-Mabrouk, Genome rearrangement, in: T. Jiang, T. Smith, Y. Xu, M.Q. Zhang (Eds.), Current Topics in Computational Molecular Biology, MIT Press, 2002]. We obtain a complete solution of the above problems for the transposition Cayley graph on the symmetric group.