The reconstruction of planar graphs

Abstract

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.