Related papers: Rigidity condition for gluing two bar-joint rigid …
The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph $G$ from some class $K$ of "pattern graphs" can be embedded into a given graph $H$ (that is, is isomorphic to a…
When all non-edge distances of a graph realized in $\mathbb{R}^{d}$ as a {\em bar-and-joint framework} are generically {\em implied} by the bar (edge) lengths, the graph is said to be {\em rigid} in $\mathbb{R}^{d}$. For $d=3$,…
We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning…
We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…
This paper treats the problem of the merging of formations, where the underlying model of a formation is graphical. We first analyze the rigidity and persistence of meta-formations, which are formations obtained by connecting several rigid…
This paper provides a combinatorial characterisation for generic forced symmetric rigidity of bar-joint frameworks in the Euclidean plane that are symmetric with respect to the orientation-reversing wallpaper group…
This paper develops a form of finite knot theory as a diagrammatic sequel to the ideal-stratum and deformation-persistence framework for knot types. Thick representatives in bounded ropelength sublevel spaces are studied through the finite…
Graphs provide an efficient tool for object representation in various computer vision applications. Once graph-based representations are constructed, an important question is how to compare graphs. This problem is often formulated as a…
This work focuses on the bearing rigidity theory, namely the branch of knowledge investigating the structural properties necessary for multi-element systems to preserve the inter-units bearings when exposed to deformations. The original…
In "Universal rigidity on the line, point orde" it is shown, answering a question of Jord\'an and Nguyen, that universal rigidity of a generic bar-joint framework in R^1 depends on more than the ordering of the vertices. The graph G that…
We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu…
Connectivity is a central notion of graph theory and plays an important role in graph algorithm design and applications. With emerging new applications in networks, a new type of graph connectivity problem has been getting more…
Given a rigid realisation of a graph $G$ in ${\mathbb R}^2$, it is an open problem to determine the maximum number of pairwise non-congruent realisations which have the same edge lengths as the given realisation. This problem can be…
We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the…
We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the $C^{d-2}_{d-1}$-cofactor rigidity of points along any (non-degenerate) conic in…
Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…
An edge cut C of a graph G is tight if |C \M| = 1 for every perfect matching M of G. Barrier-cuts and 2-separation cuts, also referred to as ELP-cuts, are two important types of tight cuts in matching covered graphs. Edmonds, Lovasz and…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…
We study the bar-and-joint frameworks in $\mathbb{R}^2$ such that some vertices are constrained to lie on some lines. The generic rigidity of such frameworks is characterised by Streinu and Theran (2010). Katoh and Tanigawa (2013) remarked…
For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings.…