Non-contractible non-edges in 2-connected graphs

Any pair of non-adjacent vertices forms a non-edge in a graph. Contraction of a non-edge merges two non-adjacent vertices into a single vertex such that the edges incident on the non-adjacent vertices are now incident on the merged vertex. In this paper, we consider simple connected graphs, hence parallel edges are removed after contraction. The minimum number of nodes whose removal disconnects the graph is the connectivity of the graph. We say a graph is k-connected, if its connectivity is k. A non-edge in a k-connected graph is contractible if its contraction does not result in a graph of lower connectivity. Otherwise the non-edge is non-contractible. We focus our study on non-contractible non-edges in 2-connected graphs. We show that cycles are the only 2-connected graphs in which every non-edge is non-contractible.     

A new force-directed graph drawing method based on edge–edge repulsion

The conventional force-directed methods for drawing undirected graphs are based on either vertex–vertex repulsion or vertex–edge repulsion. In this paper, we propose a new force-directed method based on edge–edge repulsion to draw graphs. In our framework, edges are modelled as charged springs, and a final drawing can be generated by adjusting positions of vertices according to spring forces and the repulsive forces, derived from potential fields, among edges. Different from the previous methods, our new framework has the advantage of overcoming the problem of zero angular resolution, guaranteeing the absence of any overlapping of edges incident to the common vertex. Given graph layouts probably generated by previous algorithms as the inputs to our algorithm, experimental results reveal that our approach produces promising drawings not only preserving the original properties of a high degree of symmetry and uniform edge length, but also preventing zero angular resolution and usually having larger average angular resolution. However, it should be noted that exhibiting a higher degree of symmetry and larger average angular resolution does not come without a price, as the new approach might result in the increase in undesirable overlapping of vertices as some of our experimental results indicate. To ease the problem of node overlapping, we also consider a hybrid approach which takes into account both edge–edge and vertex–vertex repulsive forces in drawing a graph.     

On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian

We consider the set G consisting of graphs of fixed order and weighted edges. The vertex set of graphs in G will correspond to point masses and the weight for an edge between two vertices is a functional of the distance between them. We pose the problem of finding the best vertex positional configuration in the presence of an additional proximity constraint, in the sense that, the second smallest eigenvalue of the corresponding graph Laplacian is maximized. In many recent applications of algebraic graph theory in systems and control, the second smallest eigenvalue of Laplacian has emerged as a critical parameter that influences the stability and robustness properties of dynamic systems that operate over an information network. Our motivation in the present work is to "assign" this Laplacian eigenvalue when relative positions of various elements dictate the interconnection of the underlying weighted graph. In this venue, one would then be able to "synthesize" information graphs that have desirable system theoretic properties.     

An optimal algorithm for solving the searchlight guarding problem on weighted two-terminal series-parallel graphs

In this paper, we consider a graph problem on a connected weighted undirected graph, called the searchlight guarding problem. Our problem is an extension of so-called graph searching/guarding problem by considering the time slot parameter in addition to the traditional building cost. Suppose that there is a fugitive who moves along the edges of the graph at any speed. We want to place a set of searchlights at the vertices to search the edges of the graph and capture the fugitive. It costs some building cost to place a searchlight at some vertex. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total costs of the vertices in S is minimized. If there is more than one set of searchlights with the minimum building cost, then find the one with the minimum searching time, that is, the time slots needed to capture the fugitive is the minimum. The problem is known to be NP-hard on weighted bipartite graphs, split graphs, and chordal graphs; and it is linear time solvable on weighted trees and interval graphs. In this paper, an algorithm is designed to solve the problem on weighted two-terminal series-parallel graphs. It works on the parsing tree structure of the given two-terminal series-parallel graph. The algorithm is divided into two phases. In the phase one, we first extract some useful properties of optimal solutions. Employing these properties, an algorithm is designed to find the set of searchlights with the minimum guarding cost and to assign the searching directions of all edges by the dynamic programming strategy. In the phase two, the searched time slots of all edges are determined by the breadth-first-search from the root of the parsing tree. The time complexities of both phases are linear. Thus, our algorithm is time optimal. Received: 12 March 1996 / 27 May 1997     

On the reachability of a version of graph-rewriting system

In this paper, we consider a problem on the reachability of a version of graph-rewriting system. It deals with 3-regular graphs with states for the vertices. They differ from ordinary graphs so that a cyclic order of the edges is assigned on each vertex. Graphs are rewritten with a rule set of graph rewriting. For any two such connected graphs with at least four vertices of distinct states, we show that there exists a rule set that rewrites one to the other.     

A Tight Lower Bound for a Special Case of Quadratic 0–1 Programming

We consider a still NP-complete partial case of the unconstrained binary quadratic optimization problem that can be described in terms of an undirected graph with red edges having negative weights and green edges having positive weights. The maximum vertex degree of the graph is three. It can be assumed w.l.o.g. that every vertex is incident to a red and a green edge. We are looking for a vertex cover with respect to the red edges which covers a subset of green edges of total weight as small as possible. We prove that for all connected such graphs except a subclass of special graphs having exactly five green edges it is possible to find a vertex cover with respect to the red edges for which the total weight of uncovered green edges is at least 1/4 fraction of the total weight of all green edges.     

Building k-connected neighborhood graphs for isometric data embedding

Isometric data embedding using geodesic distance requires the construction of a connected neighborhood graph so that the geodesic distance between every pair of data points can be estimated. This paper proposes an approach for constructing k-connected neighborhood graphs. The approach works by applying a greedy algorithm to add each edge, in a nondecreasing order of edge length, to a neighborhood graph if end vertices of the edge are not yet k-connected on the graph. The k-connectedness between vertices is tested using a network flow technique by assigning every vertex a unit flow capacity. This approach is applicable to a wide range of data. Experiments show that it gives better estimation of geodesic distances than other approaches, especially when the data are undersampled or nonuniformly distributed.     

Multiplicative vertex-colouring weightings of graphs

In this paper we conjecture that the edges of any non-trivial graph can be weighted with integers 1, 2, 3 in such a way that for every edge uv the product of weights of the edges adjacent to u is different than the product of weights of the edges adjacent to v. It is proven here for cycles, paths, complete graphs and 3-colourable graphs. It is also shown that the edges of every non-trivial graph can be weighted with integers 1, 2, 3, 4 in such a way that the adjacent vertices have different products of incident edge weights.In a total weighting of a simple graph G we assign the positive integers to edges and to vertices of G. We consider a colouring of G obtained by assigning to each vertex v the product of its weight and the weights of its adjacent edges. The paper conjectures that we can get the proper colouring in this way using the weights 1, 2 for every simple graph. We show that we can do it using the weights 1, 2, 4 on edges and 1, 2 on vertices.     

Semidynamic Algorithms for Maintaining Single-Source Shortest Path Trees

We consider the problem of updating a single-source shortest path in either a directed or an undirected graph, with positive real edge weights. Our algorithms for the incremental problem (handling edge insertions and cost decrements) work for any graph; they have optimal space requirements and query time, but their performances depend on the class of the considered graph. The cost of updates is computed in terms of amortized complexity and depends on the size of the output modifications. In the case of graphs with bounded genus (including planar graphs), graphs with bounded arboricity (including bounded degree graphs), and graphs with bounded treewidth, the incremental algorithms require O(log n) amortized time per vertex update, where a vertex is considered updated if it reduces its distance from the source. For general graphs with n vertices and m edges our incremental solution requires O( log n) amortized time per vertex update. We also consider the decremental problem for planar graphs, providing algorithms and data structures with analogous performances. The algorithms, based on Dijkstra's technique [6], require simple data structures that are really suitable for a practical and straightforward implementation. Received January 1995; revised February 1997.     

相交双圈图的代数连通度的排序

简单连通图若边数等于顶点数加1,且图中所含的两个圈至少有两个公共顶点,则称该图为相交双圈图。主要给出了相交双圈图中第五到第十大代数连通度的图类。     

Visualizing a Sequence of a Thousand Graphs (or Even More)

The visualization of dynamic graphs demands visually encoding at least three major data dimensions: vertices, edges, and time steps. Many of the state‐of‐the‐art techniques can show an overview of vertices and edges but lack a data‐scalable visual representation of the time aspect. In this paper, we address the problem of displaying dynamic graphs with a thousand or more time steps. Our proposed interleaved parallel edge splatting technique uses a time‐to‐space mapping and shows the complete dynamic graph in a static visualization. It provides an overview of all data dimensions, allowing for visually detecting time‐varying data patterns; hence, it serves as a starting point for further data exploration. By applying clustering and ordering techniques on the vertices, edge splatting on the links, and a dense time‐to‐space mapping, our approach becomes visually scalable in all three dynamic graph data dimensions. We illustrate the usefulness of our technique by applying it to call graphs and US domestic flight data with several hundred vertices, several thousand edges, and more than a thousand time steps.     

Approximation Algorithms for Intersection Graphs

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.     

An optimal algorithm for solving the searchlight guarding problem on weighted interval graphs

In this paper, a graph problem on connected, weighted, undirected graphs, called the searchlight guarding problem, is considered. Assume that there is a fugitive who moves along the edges of the graph at a random speed. The task involves placing a set of searchlights at vertices to search the edges of the graph and to spot the fugitive. Suppose that placing a searchlight at some vertex incurs some building cost. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total cost of the vertices in S is minimized. If there is more than one set of searchlights, each with a minimum building cost, then identify the set with the minimum search time, that is, where the time slots needed to spot the fugitive is the minimum. As is well established, the problem is NP-hard on weighted bipartite graphs but is linear-time solvable on weighted trees. In this paper, the design of a linear-time optimal algorithm for the searchlight guarding problem on weighted interval graphs is presented. It entails two phases. In the first phase, a set of searchlights with minimum guarding cost is identified and the search directions of all edges are assigned. To achieve this task, a new problem, called the edge-direction assignment problem, is first defined and the problem on weighted complete-split graphs is solved by the greedy strategy. Based on this computational result, the problem of finding the set of searchlights with minimum guarding cost and assigning the search directions of all edges is solved by the dynamic programming strategy. Then, in the second phase, the search time slots of each edge are determined on the basis of the results of the first phase and on some properties of interval graphs.     

An optimal graph theoretic approach to data clustering: theory andits application to image segmentation

A novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated. The data to be clustered are represented by an undirected adjacency graph 𝒢 with arc capacities assigned to reflect the similarity between the linked vertices. Clustering is achieved by removing arcs of 𝒢 to form mutually exclusive subgraphs such that the largest inter-subgraph maximum flow is minimized. For graphs of moderate size (~ 2000 vertices), the optimal solution is obtained through partitioning a flow and cut equivalent tree of 𝒢, which can be efficiently constructed using the Gomory-Hu algorithm (1961). However for larger graphs this approach is impractical. New theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size. This algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices. The new clustering algorithm is applied to the image segmentation problem. The segmentation is achieved by effectively searching for closed contours of edge elements (equivalent to minimum cuts in 𝒢), which consist mostly of strong edges, while rejecting contours containing isolated strong edges. This method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours     

Szegedy’s quantum walk with queries

When searching for a marked vertex in a graph, Szegedy’s usual search operator is defined by using the transition probability matrix of the random walk with absorbing barriers at the marked vertices. Instead of using this operator, we analyze searching with Szegedy’s quantum walk by using reflections around the marked vertices, that is, the standard form of quantum query. We show we can boost the probability to 1 of finding a marked vertex in the complete graph. Numerical simulations suggest that the success probability can be improved for other graphs, like the two-dimensional grid. We also prove that, for a certain class of graphs, we can express Szegedy’s search operator, obtained from the absorbing walk, using the standard query model.     

Pursuit-Evasion on Trees by Robot Teams

We present Graph-Clear: a novel pursuit-evasion problem on graphs which models the detection of intruders in complex indoor environments by robot teams. The environment is represented by a graph, and a robot team can execute sweep and block actions on vertices and edges, respectively. A sweep action detects intruders in a vertex and represents the capability of the robot team to detect intruders in the region associated to the vertex. Similarly, a block action prevents intruders from crossing an edge and represents the capability to detect intruders as they move between regions. Both actions may require multiple robots to be executed. A strategy is a sequence of block and sweep actions to detect all intruders. When instances of Graph-Clear are being solved, the goal is to determine optimal strategies, i.e., strategies that use the least number of robots. We prove that for the general case of graphs, the problem of computation of optimal strategies is NP-hard. Next, for the special case of trees, we provide a polynomial-time algorithm. The algorithm ensures that throughout the execution of the strategy, all cleared vertices form a connected subtree, and we show that it produces optimal strategies.      

SOLVING ABSTRACT COOPERATIVE PATH‐FINDING IN DENSELY POPULATED ENVIRONMENTS

The problem of cooperative path‐finding is addressed in this work. A set of agents moving in a certain environment is given. Each agent needs to reach a given goal location. The task is to find spatial temporal paths for agents such that they eventually reach their goals by following these paths without colliding with each other. An abstraction where the environment is modeled as an undirected graph is adopted—vertices represent locations and edges represent passable regions. Agents are modeled as elements placed in the vertices while at most one agent can be located in a vertex at a time. At least one vertex remains unoccupied to allow agents to move. An agent can move into unoccupied neighboring vertex or into a vertex being currently vacated if a certain additional condition is satisfied. Two novel scalable algorithms for solving cooperative path‐finding in bi‐connected graphs are presented. Both algorithms target environments that are densely populated by agents. A theoretical and experimental evaluation shows that the suggested algorithms represent a viable alternative to search based techniques as well as to techniques employing permutation groups on the studied class of the problem.     

Algorithms for Partition of Some Class of Graphs under Compaction and Vertex-Compaction

The compaction problem is to partition the vertices of an input graph G onto the vertices of a fixed target graph H, such that adjacent vertices of G remain adjacent in H, and every vertex and non-loop edge of H is covered by some vertex and edge of G respectively, i.e., the partition is a homomorphism of G onto H (except the loop edges). Various computational complexity results, including both NP-completeness and polynomial time solvability, have been presented earlier for this problem for various classes of target graphs H. In this paper, we pay attention to the input graphs G, and present polynomial time algorithms for the problem for some class of input graphs, keeping the target graph H general as any reflexive or irreflexive graph. Our algorithms also give insight as for which instances of the input graphs, the problem could possibly be NP-complete for certain target graphs. With the help of our results, we are able to further refine the structure of the input graph that would be necessary for the problem to be possibly NP-complete, when the target graph is a cycle. Thus, when the target graph is a cycle, we enhance the class of input graphs for which the problem is polynomial time solvable. We also present analogous results for a variation of the compaction problem, which we call the vertex-compaction problem. Using our results, we also provide important relationships between compaction, retraction, and vertex-compaction to cycles.     

Using partial evaluation in holistic subgraph search

Because of its wide application, the subgraph matching problem has been studied extensively during the past decade. However, most existing solutions assume that a data graph is a vertex/edge-labeled graph (i.e., each vertex/edge has a simple label). These solutions build structural indices by considering the vertex labels. However, some real graphs contain rich-content vertices such as user profiles in social networks and HTML pages on the World Wide Web. In this study, we consider the problem of subgraph matching using a more general scenario. We build a structural index that does not depend on any vertex content. Based on the index, we design a holistic subgraph matching algorithm that considers the query graph as a whole and finds one match at a time. In order to further improve efficiency, we propose a “partial evaluation and assembly” framework to find subgraph matches over large graphs. Last but not least, our index has light maintenance overhead. Therefore, our method can work well on dynamic graphs. Extensive experiments on real graphs show that our method outperforms the state-of-the-art algorithms.     

Experimental Analysis of Heuristic Algorithms for the Dominating Set Problem

We say a vertex v in a graph G covers a vertex w if v=w or if v and w are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy for finding a small dominating set in a graph consists of successively choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations is presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments. Received October 27, 1998; revised March 25, 2001.