Uniformly dissociated graphs

Creative Commons Attribution 3.0 International LicenseA set D of vertices in a graph G is called a dissociation set if every vertex in D has at most one neighbor in D. We call a graph G uniformly dissociated if all maximal dissociation sets are of the same cardinality. Characterizations of uniformly dissociated graphs with small cardinalities of dissociation sets are proven; in particular, the graphs in which all maximal dissociation sets are of cardinality 2 are the complete graphs on at least two vertices from which possibly a matching is removed, while the graphs in which all maximal dissociation sets are of cardinality 3 are the complements of the K4-free geodetic graphs with diameter 2. A general construction by which any graph can be embedded as an induced sub graph of a uniformly dissociated graph is also presented. In the main result we characterize uniformly dissociated graphs with girth at least 7 to be either isomorphic to C7, or obtainable from an arbitrary graph H with girth at least 7 by identifying each vertex of H with a leaf of a copy of P3

Drawing Shortest Paths in Geodetic Graphs

Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph $G$, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of $G$, i.e., a drawing of $G$ in which the curves of any two shortest paths meet at most once? We answer this question in the negative by showing the existence of geodetic graphs that require some pair of shortest paths to cross at least four times. The bound on the number of crossings is tight for the class of graphs we construct. Furthermore, we exhibit geodetic graphs of diameter two that do not admit a philogeodetic drawing.Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020

Hyperbolicity vs. amenability for planar graphs

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps

Statistical testing of directions observations independence

Independence of observations is often assumed when adjusting geodetic network. Unlike the\ud distance observations, no dependence of environmental conditions is known for horizontal\ud direction observations. In order to determine the dependence of horizontal direction observations,\ud we established test geodetic network of a station and four observation points. Measurements of\ud the highest possible accuracy were carried out using Leica TS30 total station along with precise\ud prisms GPH1P. Two series of hundred sets of angles were measured, with the first one in bad\ud observation conditions. Using different methods, i.e. variance–covariance matrices, x2 test and analyses of time series, the independence of measured directions, reduced directions and horizontal angles were tested. The results show that the independence of horizontal direction\ud observations is not obvious and certainly not in poor conditions. In this case, it would be appropriate for geodetic network adjustments to use variance–covariance matrix calculated from measurements instead of diagonal variance–covariance matrix

A characterization of bipartite graphs associated with BIB-designs with λ = 1

AbstractA graph is said to be F-geodetic (for some function F) if the number of shortest paths between two vertices at distance i is F(i). It is shown that a bipartite F-geodetic graph with diameter ⩽4 is either (i)a tree, or(ii)a distance-regular graph, or(iii)the graph associated with a BIB-design with λ = 1

An analytical model of layered continuous beams with partial interaction

Starting with the geometrically non-linear formulation and the subsequent linearization, this paper presents a consistent formulation of the exact mechanical analysis of geometrically and materially linear three-layer continuous planar beams. Each layer of the beam is described by the geometrically linear beam theory. Constitutive laws of layer materials and relationships between interlayer slips and shear stresses at the interface are assumed to be linear elastic. The formulation is first applied in the analysis of a three-layer simply supported beam. The results are compared to those of Goodman and Popov (1968) and to those obtained from the formulation of the European code for timber structures, Eurocode 5 (1993). Comparisons show that the present and the Goodman and Popov (1968) results agree completely, while the Eurocode 5 (1993) results differ to a certain degree. Next, the analytical solution is used in formulating a general procedure for the analysis of layered continuous beams. The applications show the qualitative and quantitative effects of the layer and the interlayer slip stiffnesses on internal forces, stresses and deflections of composite continuous beams

Analytical solution of two-layer beam taking into account interlayer slip and shear deformation

A mathematical model is proposed and its analytical solution derived for the analysis of the geometrically and materially linear two-layer beams with different material and geometric characteristics of an individual layer. The model takes into account the effect of the transverse shear deformation on displacements in each layer. The analytical study is carried out to evaluate the influence of the transverse shear deformation on the static and kinematic quantities. We study a simply supported two-layer planar beam subjected to the uniformly distributed load. Parametric studies have been performed to investigate the influence of shear by varying material and geometric parameters, such as interlayer slip modulus (K), flexural-to-shear moduli ratios (E/G) and span-to-depth ratios (L/h). The comparison of the results for vertical deflections shows that shear deformations are more important for high slip modulus, for ``short'' beams with small L/h ratios, and beams with high E/G ratios. In these cases, the effect of the shear deformations becomes significant and has to be addressed in design. It also becomes apparent that models, which consider the partial interaction between the layers, should be employed if beams have very flexible connections