Vertex-edge Graph Research Articles

On the dimer problem of the vertex-edge graph of a cubic graph

Let G be a graph with vertex set V(G) and edge set E(G), and L(G) be the line graph of G, which has vertex set V(L(G))=E(G) and two vertices e and f in L(G) are adjacent if and only if two edges e and f in G have a common vertex. The vertex-edge graph M(G) of G has vertex set V(G)∪E(G) and edge set E(L(G))∪{ue,ve|∀e=uv∈E(G)}. In this paper, we show that if G is a connected cubic graph with an even number of edges, then the number of dimer coverings of M(G) equals 2|V(G)|/2+13|V(G)|/4. As an application, we obtain the exact solution of the dimer problem of the weighted silicate network obtained from the hexagonal lattice in the context of statistical physics.

Integrating vertex and edge features with Graph Convolutional Networks for skeleton-based action recognition

Methods based on Graph Convolutional Networks(GCN) for skeleton-based action recognition have achieved great success due to their ability to exploit graph structural information from skeleton data. Recently, the bone information has attracted considerable attention as an effective modality which complements the more conventional joint information for action recognition. However, most existing GCN-based methods extract the joint and bone features with two separate GCN networks, ignoring the dependencies between them. In this paper, a novel GCN model is proposed to exploit the information across joints, bones and their relationship collaboratively on a single undirected graph instead of two separate networks. We call the proposed model Vertex-Edge Graph Convolutional Network (VE-GCN) since it conducts the graph convolution operation on the sampling area containing the designated vertexes from joints and edges from bones, respectively. In addition to conducting the Vertex-Edge graph convolution based on the physical connections of the skeleton, we further apply the Vertex-Edge graph convolution to the non-physical joint-joint and joint-bone connections to capture the distal dependencies, and then the convolution results on the non-physical connections are incorporated into the VE-GCN. Moreover, the Conditional Random Field (CRF) is adopted as the loss function to achieve the task of action recognition. Experimental results on four challenging benchmarks (NTU RGB+D, NTU RGB+D 120, N-UCLA, SYSU) show that the proposed model achieves state-of-the-art performance.

Motion planning and control of a planar polygonal linkage

For a polygonal linkage, we produce a fast navigation algorithm on its configuration space. The basic idea is to approximate M(L) by the vertex-edge graph of the cell decomposition of the configuration space discovered by the first author. The algorithm has three aspects: (1) the number of navigation steps does not exceed 15 (independent of the linkage), (2) each step is a disguised flex of a quadrilateral from one triangular configuration to another, which is a well understood type of flex, and (3) each step can be performed explicitly by adding some extra bars and obtaining a mechanism with one degree of freedom.

More bounds on the diameters of convex polytopes

Let Δ(d, n) be the maximum possible diameter of the vertex-edge graph over all d-dimensional polytopes defined by n inequalities. The Hirsch bound holds for particular n and d if Δ(d, n)≤n−d. Francisco Santos recently resolved a question open for more than five decades by showing that Δ(d, 2d)≥d+1 for d=43; the dimension was then lowered to 20 by Matschke, Santos and Weibel. This progress has stimulated interest in related questions. The existence of a polynomial upper bound for Δ(d, n) is still an open question, the best bound being the quasi-polynomial one due to Kalai and Kleitman in 1992. Another natural question is for how large n and d the Hirsch bound holds. Goodey showed in 1972 that Δ(4, 10)=5 and Δ(5, 11)=6, and more recently, Bremner and Schewe showed that Δ(4, 11)=Δ(6, 12)=6. Here, we show that Δ(4, 12)=Δ(5, 12)=7.

Optimal pants decompositions and shortest homotopic cycles on an orientable surface

We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solve this problem in a framework where the cycles are closed walks on the vertex-edge graph of a combinatorial surface that may overlap but do not cross. We give an algorithm that transforms an input pants decomposition into another homotopic pants decomposition that is optimal : each cycle is as short as possible in its homotopy class. As a consequence, finding a shortest cycle homotopic to a given simple cycle amounts to extending the cycle into a pants decomposition and to optimizing it: the resulting pants decomposition contains the desired cycle. We describe two algorithms for extending a cycle to a pants decomposition. All algorithms in this article are polynomial, assuming uniformity of the weights of the vertex-edge graph of the surface.

Optimal System of Loops on an Orientable Surface

Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a {\it system of loops} for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.

Activities: Massive Graphs, Power Laws, and the World Wide Web

Mathematical modeling has seen many changes over the years. These changes range from the types of situations being modeled to the types of tools used for the modeling. An extremely powerful modeling tool for many situations is the vertex-edge graph (hereafter simply called a graph). In this type of graph, a set of points, called vertices (or nodes), represent objects, people, or other ideas. The nodes can be connected with (not necessarily straight) line segments, called edges (or arcs), to show a relationship between the nodes. Graphs are used to model everything from transportation networks to groups of friends.

Three-dimensional shape pattern recognition using vertex classification and vertex-edge graphs

A method for computer recognition of shape patterns from a three-dimensional (3D) boundary representation of a solid object is described. The vertices of the object in the solid modelling database are classified by analysing the topology and geometric properties surrounding the vertices. Using vertex types to label the nodes of a vertex-edge (V-E) graph of a designed object, a labelled graph with sufficient shape information for recognition is established. The graphs for regional shape patterns can be defined in the forms of these labelled graphs. The recognition of each single regional pattern requires matching the pattern graph to subgraphs embedded in the labelled graph for the object. A hierarchy can be established for shape patterns identified by this method. The role of pattern recognition for CAD/CAM integration has been discussed in this research.