On the 2-sum in rigidity matroids

AbstractWe show that the graph 2-sum of two frameworks is the underlying framework for the 2-sum of the infinitesimal and generic rigidity matroids of the frameworks. However, we show that, unlike the cycle matroid of a graph, these rigidity matroids are not closed under 2-sum decomposition

A polynomial time algorithm for determining zero Euler–Petrie genus of an Eulerian graph

AbstractA dual-Eulerian graph is a plane graph which has an ordering defined on its edge set which forms simultaneously an Euler circuit in the graph and an Euler circuit in the dual graph. Dual-Eulerian graphs were defined and studied in the context of silicon optimization of cmos layouts. They are necessarily of low connectivity, hence may have many planar embeddings. We give a polynomial time algorithm to answer the question whether or not a planar multigraph admits an embedding which is dual-Eulerian and construct such an embedding, if it exists

Non-crossing frameworks with non-crossing reciprocals

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that if such an embedding is sufficiently generic, then the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation embeddings of Laman circuits. All self-stresses on a planar graph correspond to liftings to piece-wise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure

Single crystal growth of BaFe2−x_{2-x}Cox_xAs2_2 without fluxing agent

We report a simple, reliable method to grow high quality BaFe$_{2-x}$Co$_x$As$_2$ single crystal samples without using any fluxing agent. The starting materials for the single crystal growth come from well-crystallized polycrystalline samples and the highest growing temperature can be 1493 K. The as-grown crystals have typical dimensions of 4$\times3\times$0.5 mm$^3$ with c-axis perpendicular to the shining surface. We find that the samples have very large current carrying ability, indicating that the samples have good potential technological applications.Comment: 8 pages, 4 figures, accepted by Journal of Superconductivity and Novel Magnetis

Equilibrium stressability of multidimensional frameworks

We prove an equilibrium stressability criterium for trivalent multidimensional tensegrities. The criterium appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1-forms, and (4) in Cayley algebra terms

The 24 symmetry pairings of self-dual maps on the sphere

Abstract. Given a self–dual map on the sphere, the collection of its self– dual permutations generates a transformation group in which the map automorphism group appears as a subgroup of index two. A careful examination of this pairing yields direct constructions of self–dual maps and provides a classification of self–dual maps. 1

Self-dual graphs

Abstract. We consider the three forms of self-duality that can be exhibited by a planar graph G, map self-duality, graph self-duality and matroid selfduality. We show how these concepts are related with each other and with the connectivity of G. We use the geometry of self-dual polyhedra together with the structure of the cycle matroid to construct all self-dual graphs. 1. Self-Duality of Graphs 1.1. Forms of Self-duality. Given a planar graph G = (V, E), any regular embedding of the topological realization of G into the sphere partitions the sphere into regions called the faces of the embedding, and we write the embedded graph, called a map, as M = (V, E, F). G may have loops and parallel edges. Given a map M, we form the dual map, M ∗ by placing a vertex f ∗ in the center of each face f, and for each edge e of M bounding two faces f1 and f2, we draw a dual edge e ∗ connecting the vertices f ∗ 1 and f ∗ 2 and crossing e once transversely. Each vertex v of M will then correspond to a face v ∗ of M ∗ and we write M ∗ = (F ∗ , E ∗ , V ∗). If the graph G has distinguishable embeddings, then G may have more than one dual graph, see Figure 1. In this example a portion of the map (V, E, F) is flipped over on

Rigidity, global rigidity, and graph decomposition

AbstractThe recent combinatorial characterization of generic global rigidity in the plane by Jackson and Jordán (2005) [10] recalls the vital relationship between connectivity and rigidity that was first pointed out by Lovász and Yemini (1982) [13]. The Lovász–Yemini result states that every 6-connected graph is generically rigid in the plane, while the Jackson–Jordán result states that a graph is generically globally rigid in the plane if and only if it is 3-connected and edge-2-rigid.We examine the interplay between the connectivity properties of the connectivity matroid and the rigidity matroid of a graph and derive a number of structure theorems in this setting, some well known, some new. As a by-product we show that the class of generic rigidity matroids is not closed under 2-sum decomposition. Finally we define the configuration index of the graph and show how the structure theorems can be used to compute it