Thesis (Ph. D.)--West Virginia University, 2009. / Title from document title page. Document formatted into pages; contains vi, 84 p. : ill. Includes abstract. Includes bibliographical references (p. 71-74).

Thesis (M.S.)--The University of North Carolina at Greensboro, 2008. / Directed by Paul Duvall; submitted to the Dept. of Mathematical Sciences. Title from PDF t.p. (viewed Sep. 4, 2009). Includes bibliographical references (p. 57).

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Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 52 p. : ill. Includes abstract. Includes bibliographical references (p. 50-52).

Given a finite set S of n nonintersecting line segments with no three end points collinear, the segment end point visibility graph is defined as the graph whose vertices are the end points of the line segments in S and two vertices are adjacent if the straight line segment joining two end points does not intersect any element of S, or if they are end points of the same segment. Segment end point visibility graphs have a wide variety of applications in VLSI circuit design, study of art gallery problems, and other areas of computational geometry. This thesis contains a survey of the important results that are currently known regarding the characterization of these graphs. Also a weak visibility dual of a segment end point visibility graph is defined and some structural properties of such graphs are presented. Some open problems and questions related to the characterization of weak visibility graphs are also discussed. / Department of Mathematical Sciences

Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2008/05/29). Includes bibliographical references.

Interval graphs admit elegant ordering and structural characterizations. A natural digraph analogue of interval graphs, called chronological interval digraphs, has recently been identified and studied. We introduce the class of chronological rectangle digraphs, and show that they are a higher dimensional analogue of chronological interval digraphs. A main goal of this thesis is to establish a foundation of knowledge about this class, including basic properties and an ordering characterization. Our most significant result is a forbidden induced subdigraph characterization for the series-parallel digraphs which are chronological rectangle. We also discuss obtaining chronological rectangle digraphs from orientations of graphs. In addition we introduce the related concept of the chronological interval dimension of a digraph, and determine the digraphs for which it is defined. Unit and proper chronological rectangle digraphs, defined analogously to unit and proper interval graphs, are also introduced and studied. / Graduate

Analyzes the results of M.I. Ostrovskii's theorem of inequalities which estimate the minimal edge congestion for finite simple graphs. Uses the generic results of the theorem to examine and further reduce the parameters of inequalities for specific families of graphs, particularly complete graphs and complete bipartite graphs. Also, explores a possible minimal congestion tree for some grids while forming a conjecture for all grids.

Chapter I begins with a brief history of the topic of greatest common subgraphs. Then we provide a summaiy of the work done on some variations of greatest common subgraphs. Finally, in this chapter we present results previously obtained on greatest common divisors and least common multiples of graphs. In Chapter II the concepts of prime graphs, prime divisors of graphs, and primeconnected graphs are presented. We show the existence of prime trees of any odd size and the existence of prime-connected trees that are not prime having any odd composite size. Then the number of prime divisors in a graph is studied. Finally, we present several results involving the existence of graphs whose size satisfies some prescribed condition and which contains a specified number of prime divisors. Chapter III presents properties of greatest common divisors and least common multiples of graphs. Then graphs with a prescribed number of greatest common divisors or least common multiples are studied. In Chapter IV we study the sizes of greatest common divisors and least common multiples of specified graphs. We find the sizes of greatest common divisors and least common multiples of stars and that of stripes. Then the size of greatest common divisors and least common multiples of paths and complete graphs are investigated. In particular, the size of least common multiples of paths versus K3 or K4 are determined. Then we present the greatest common divisor index of a graph and we determine this parameter for several classes of graphs. iii In Chapter V greatest common divisors and least common multiples of digraphs are introduced. The existence of least common mutliples of two stars is established, and the size of a least common multiple is found for several pairs of stars. Finally, we present the concept of greatest common divisor index of a digraph and determine it for several classes of digraphs. iv / Mathematical Sciences / Ph. D. (Mathematical sciences)

published_or_final_version / Electrical Engineering / Master / Master of Philosophy