A Proof of Tutte's Realizability Condition

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB 07-67-C-019

Discrete events: Perspectives from system theory

Systems Theory;differentiaal/ integraal-vergelijkingen

On Geometric Drawings of Graphs

This thesis is about geometric drawings of graphs and their topological generalizations. First, we study pseudolinear drawings of graphs in the plane. A pseudolinear drawing is one in which every edge can be extended into an infinite simple arc in the plane, homeomorphic to $\mathbb{R}$, and such that every two extending arcs cross exactly once. This is a natural generalization of the well-studied class of rectilinear drawings, where edges are straight-line segments. Although, the problem of deciding whether a drawing is homeomorphic to a rectilinear drawing is NP-hard, in this work we characterize the minimal forbidden subdrawings for pseudolinear drawings and we also provide a polynomial-time algorithm for recognizing this family of drawings. Second, we consider the problem of transforming a topological drawing into a similar rectilinear drawing preserving the set of crossing pairs of edges. We show that, under some circumstances, pseudolinearity is a necessary and sufficient condition for the existence of such transformation. For this, we prove a generalization of Tutte's Spring Theorem for drawings with crossings placed in a particular way. Lastly, we study drawings of $K_n$ in the sphere whose edges can be extended to an arrangement of pseudocircles. An arrangement of pseudocircles is a set of simple closed curves in the sphere such that every two intersect at most twice. We show that (i) there is drawing of $K_{10}$ that cannot be extended into an arrangement of pseudocircles; and (ii) there is a drawing of $K_9$ that can be extended to an arrangement of pseudocircles, but no extension satisfies that every two pseudocircles intersects exactly twice. We also introduce the notion pseudospherical drawings of $K_n$, a generalization of spherical drawings in which each edge is a minor arc of a great circle. We show that these drawings are characterized by a simple local property. We also show that every pseudospherical drawing has an extension into an arrangement of pseudocircles where the ``at most twice'' condition is replaced by ``exactly twice''

Graph theory in America 1876-1950

This narrative is a history of the contributions made to graph theory in the United States of America by American mathematicians and others who supported the growth of scholarship in that country, between the years 1876 and 1950. The beginning of this period coincided with the opening of the first research university in the United States of America, The Johns Hopkins University (although undergraduates were also taught), providing the facilities and impetus for the development of new ideas. The hiring, from England, of one of the foremost mathematicians of the time provided the necessary motivation for research and development for a new generation of American scholars. In addition, it was at this time that home-grown research mathematicians were first coming to prominence. At the beginning of the twentieth century European interest in graph theory, and to some extent the four-colour problem, began to wane. Over three decades, American mathematicians took up this field of study - notably, Oswald Veblen, George Birkhoff, Philip Franklin, and Hassler Whitney. It is necessary to stress that these four mathematicians and all the other scholars mentioned in this history were not just graph theorists but worked in many other disciplines. Indeed, they not only made significant contributions to diverse fields but, in some cases, they created those fields themselves and set the standards for others to follow. Moreover, whilst they made considerable contributions to graph theory in general, two of them developed important ideas in connection with the four-colour problem. Grounded in a paper by Alfred Bray Kempe that was notorious for its fallacious 'proof' of the four-colour theorem, these ideas were the concepts of an unavoidable set and a reducible configuration. To place the story of these scholars within the history of mathematics, America, and graph theory, brief accounts are presented of the early years of graph theory, the early years of mathematics and graph theory in the USA, and the effects of the founding of the first institute for postgraduate study in America. Additionally, information has been included on other influences by such global events as the two world wars, the depression, the influx of European scholars into the United States of America, mainly during the 1930s, and the parallel development of graph theory in Europe. Until the end of the nineteenth century, graph theory had been almost entirely the prerogative of European mathematicians. Perhaps the first work in graph theory carried out in America was by Charles Sanders Peirce, arguably America's greatest logician and philosopher at the time. In the 1860s, he studied the four-colour conjecture and claimed to have written at least two papers on the subject during that decade, but unfortunately neither of these has survived. William Edward Story entered the field in 1879, with unfortunate consequences, but it was not until 1897 that an American mathematician presented a lecture on the subject, albeit only to have the paper disappear. Paul Wernicke presented a lecture on the four-colour problem to the American Mathematician Society, but again the paper has not survived. However, his 1904 paper has survived and added to the story of graph theory, and particularly the four-colour conjecture. The year 1912 saw the real beginning of American graph theory with Veblen and Birkhoff publishing major contributions to the subject. It was around this time that European mathematicians appeared to lose interest in graph theory. In the period 1912 to 1950 much of the progress made in the subject was from America and by 1950 not only had the United States of America become the foremost country for mathematics, it was the leading centre for graph theory