A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we explore various aspects of closed graphs, including the number of closed labelings and clustering coefficients.

In the design activity, part geometry is assembled together to create an assembly model. The number of parts can range from a few tens to a few millions and typically the relationships among them construct a closed-loop with under-constrained states in a constraint graph. In this paper, we propose an efficient and robust 3D constraint solving method for determining part positions in closed-loop...

We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer k-planar graphs, where each edge is crossed by at most k other edges; and, outer k-quasi-planar graphs where no k edges can mutually cross. We show that the outer k-planar graphs are (b √ 4k + 1...

A graph is Gaussian if it is the graph of arcs and self-intersections of a closed C∞ curve in the plane. In this note we describe a recursive characterization of 4-regular Gaussian graphs and give conditions under which the Gaussian property is a graph invariant.

A corollary of a celebrated theorem of Minty is that the subdifferential graph of a closed convex function on Rn has uniform local dimension n. In contrast, there exist nonconvex closed functions whose subdifferentials have large graphs. We consider how far Minty’s corollary extends to functions that are nonconvex but semi-algebraic. © 2011 Elsevier Ltd. All rights reserved.

Consider a gain graph with abelian gain group having no odd torsion. If there is a binary cycle basis each of whose members can be lifted to a closed walk whose gain is the identity, then the gain graph is balanced, provided that the graph is finite or the group has no nontrivial infinitely 2-divisible elements.

The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the total weight of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle...

The present work aims to exploit the interplay between algebraic properties of rings and graph-theoretic structures some associated graphs. We introduce commutatively closed graphs investigate subsets a ring with help graph theory. In particular, we compute diameter matrix Artinian semisimple algebras.

We prove that a binomial edge ideal of a graph G has a quadratic Gröbner basis with respect to some term order if and only if the graph G is closed with respect to a given labelling of the vertices. We also state some criteria for the closedness of a graph G that do not depend on the labelling of its vertex set.

Manual derivation of the dynamic model of a modular robot is almost impossible because it may have very different geometries and DOFS through module reconfiguration. This paper presents an algorithm to automatically generate the closed-form equation of motion of a modular robot from a kinematic graph based representation of the assembly configuration. We consider modular robots with the more ge...