Related papers: Global Rigidity of Line Constrained Frameworks
We develop a purely combinatorial theory of limit linear series on metric graphs. This will be based on the formalisms of hypercube rank functions and slope structures. We provide a full classification of combinatorial limit linear series…
We present a new semidefinite Farkas lemma involving a side constraint on the rank. This lemma is then used to present a new proof of a recent characterization, by Connelly and Gortler, of dimensional rigidity of bar frameworks.
A foundational theorem of Laman provides a counting characterisation of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterisation was obtained for…
Rigidity theory enables us to specify the conditions of unique localizability in the cooperative localization problem of wireless sensor networks. This paper presents a combinatorial rigidity approach to measure (i) generic rigidity and…
A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on…
In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…
The problem of characterizing the structure of an elastic network constrained to lie on a frozen curved surface appears in many areas of science and has been addressed by many different approaches, most notably, extending linear elasticity…
A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. We establish tight lower bounds on the maximum size of a uniquely restricted matching in terms of order, size, and maximum degree.
The goal of this paper is to unify two lines in a particular area of graph limits. First, we generalize and provide unified treatment of various graph limit concepts by means of a combination of model theory and analysis. Then, as an…
A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek-Geiringer-Laman type. Methods from matroid theory have been used to prove other interesting results, again under the…
It is a famous result of Lovasz and Yemini (1982) that 6-connected graphs are rigid in the plane. This was recently improved by Jackson and Jordan (2009) who showed that 6-mixed connectivity is also sufficient for rigidity. Here we give…
In recent work by Johnson et al. (2022), a framework was described for the study of graph problems over classes specified by omitting each of a finite set of graphs as subgraphs. If a problem falls into the framework then its computational…
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…
The analysis of networks characterized by links with heterogeneous intensity or weight suffers from two long-standing problems of arbitrariness. On one hand, the definitions of topological properties introduced for binary graphs can be…
A matching $M$ in a graph $G$ is uniquely restricted if there is no matching $M'$ in $G$ that is distinct from $M$ but covers the same vertices as $M$. Solving a problem posed by Golumbic, Hirst, and Lewenstein, we characterize the graphs…
Global constraints proved themselves to be an efficient tool for modelling and solving large-scale real-life combinatorial problems. They encapsulate a set of binary constraints and using global reasoning about this set they filter the…
In this article we consider resistance matrix of a connected graph. For unweighted graph we study some necessary and sufficient conditions for resistance regular graphs. Also we find some relationship between Laplacian matrix and resistance…
We use Hanf locality and a result of Cruickshank, Jackson, and Tanigawa on the global rigidity of graphs of $k$-circuits to prove that local and global $d$-rigidity are not definable in the first order logic of graphs.
A minimally rigid graph, also called Laman graph, models a planar framework which is rigid for a general choice of distances between its vertices. In other words, there are finitely many ways, up to isometries, to realize such a graph in…
The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for…