The reconstruction of planar graphs

The object of this thesis is to investigate the Reconstruction Problem for planar graphs. This study naturally leads to related topics concerning certain nonplanar graphs and the use of their embeddings on appropriate surfaces to reconstruct them. The principal aim of this work is to find new techniques of reconstruction and to increase the number of classes of graphs known to be reconstructible. In achieving this aim, various important properties of graphs, such as connectivity and uniqueness of embeddings, are explored, and new results on these topics are obtained. Part I, which consists of three chapters, contains a historic a l, non-technical introduction and general graph-theoretical definitions, notation and results. Some new concepts in reconstruction are also presented, notably the idea of reconstructor sets. Part II of the thesis deals with the vertex-reconstruction of maximal planar graphs: Chapter 4 is concerned with the vertex-recognition of maximal planarity, whereas Chapter 5 deals with the vertex-reconstruction. Part III deals with edge-reconstruction: planar graphs with minimum valency 5 and 4-connected planar graphs are reconstructed in Chapters 6 and 7 respectively. In Chapter 7, extensive use is made of the concept of reconstructor sets introduced in Chapter 3. This chapter also contains a brief discussion on the reconstruction of graphs from edge-contracted subgraphs, a problem which, in certain cases, can be regarded as dual to the Edge-reconstruction Problem. Part IV is concerned with extending the results and techniques of the previous chapters to nonplanar graphs. Chapter 8 discusses where the previous techniques fa il, and indicates where new methods are needed. In Chapter 9, all graphs which triangulate some surface and have connectivity 3 are edge-reconstructed. Certain graphs which triangulate the torus or the projective plane are also shown to be weakly vertex-reconstructible. Chapter 10 deals with the edge-reconstruction of all graphs which triangulate the projective plane. The Appendix proves a conjecture of Harary on the cutvertex-reconstruction of trees. One technique used here ties up with a method employed in previous chapters on edge-reconstruction

The reconstruction of maximal planar graphs II. Reconstruction

AbstractIn the first paper [3], the author, together with Fiorini, has shown that maximal planar graphs are recognizable from their decks of vertex-deleted subraphs. The aim of this paper is to show that such graphs are reconstructible

Surface networks

© Copyright CASA, UCL. The desire to understand and exploit the structure of continuous surfaces is common to researchers in a range of disciplines. Few examples of the varied surfaces forming an integral part of modern subjects include terrain, population density, surface atmospheric pressure, physico-chemical surfaces, computer graphics, and metrological surfaces. The focus of the work here is a group of data structures called Surface Networks, which abstract 2-dimensional surfaces by storing only the most important (also called fundamental, critical or surface-specific) points and lines in the surfaces. Surface networks are intelligent and “natural ” data structures because they store a surface as a framework of “surface ” elements unlike the DEM or TIN data structures. This report presents an overview of the previous works and the ideas being developed by the authors of this report. The research on surface networks has fou

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automorphisms simply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications

Polytopes and Loop Quantum Gravity

The main aim of this thesis is to give a geometrical interpretation of ``spacetime grains'' at Planck scales in the framework of Loop Quantum Gravity. My work consisted in analyzing the details of the interpretation of the quanta of space in terms of polytopes. The main results I obtained are the following: We clarified details on the relation between polytopes and interwiners, and concluded that an intertwiner can be seen unambiguously as the state of a \emph{quantum polytope}. Next we analyzed the properties of these polytopes: studying how to reconstruct the solid figure from LQG variables, the possible shapes and the volume. We adapted existing algorithms to express the geometry of the polytopes in terms of the holonomy-fluxes variables of LQG, thus providing an explicit bridge between the original variables and the interpretation in terms of polytopes of the phase space. Finally we present some direct application of this geometrical picture. We defined a volume operator such as in the large spin limit it reproduce the geometrical volume of a polytope, we computed numerically his spectrum for some elementary cases and we pointed out some asymptotic property of his spectrum. We discuss applications of the picture in terms of polytopes to the study of the semiclassical limit of LQG, in particular commenting a connection between the quantum dynamics and a generalization of Regge calculus on polytopes