A note on the convexity number for complementary prisms

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.Comment: 10 pages, 2 figure

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

Leaps: an approach to the block structure of a graph

To study the block structure of a connected graph G=(V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation +G as well as the set of leaps LG of the connected graph G. The underlying graph of +G , as well as that of LG , turns out to be just the block closure of G (i.e. the graph obtained by making each block of G into a complete subgraph).

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

Advanced SBAS-DInSAR technique for controlling large civil infrastructures: an application to the Genzano di Lucania dam

Monitoring surface deformation on dams is commonly carried out by in situ geodetic surveying, which is time consuming and characterized by some limitations in space coverage and frequency. More recently microwave satellite-based technologies, such as advanced-DInSAR (Differential Synthetic Aperture Radar Interferometry), have allowed the integration and improvement of the observation capabilities of ground-based methods thanks to their effectiveness in collecting displacement measurements on many non-destructive control points, corresponding to radar reflecting targets. The availability of such a large number of points of measurement, which are distributed along the whole structure and are characterized by millimetric accuracy on displacement rates, can be profitably adopted for the calibration of numerical models. These models are implemented to simulate the structural behaviour of a dam under conditions of stress thus improving the ability to maintain safety standards. In this work, after having analysed how advanced DInSAR can effectively enhance the results from traditional monitoring systems that provide comparable accuracy measurements on a limited number of points, an FEM model of the Genzano di Lucania earth dam is developed and calibrated. This work is concentrated on the advanced DInSAR technique referred to as Small BAseline Subset (SBAS) approach, benefiting from its capability to generate deformation time series at full spatial resolution and from multi-sensor SAR data, to measure the vertical consolidation displacement of the Genzano di Lucania earth dam

Geodetic Graphs: Experiments and New Constructions

In 1962 Ore initiated the study of geodetic graphs. A graph is called geodetic if the shortest path between every pair of vertices is unique. In the subsequent years a wide range of papers appeared investigating their peculiar properties. Yet, a complete classification of geodetic graphs is out of reach. In this work we present a program enumerating all geodetic graphs of a given size. Using our program, we succeed to find all geodetic graphs with up to 25 vertices and all regular geodetic graphs with up to 32 vertices. This leads to the discovery of two new infinite families of geodetic graphs