Representations of unit-distance graphs

Boris Horvat (2009) Representations of unit-distance graphs. PhD thesis.

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Abstract

The doctoral thesis describes problems concerning graphs that can be represented in the Euclidean plane (or k-space) in such a way, that vertices are represented as points in the plane (k-space) and edges as line segments of unit lengths. Problems are observed from a computational and a mathematical point of view. In the first part of the thesis the (already known, mainly mathematical) theory of unit-distance graph representations is presented; at the same time the terminology of the results is unified and several propositions are proved. First computer aided attempts to generate small graphs with a unit-distance representation are discussed. In the following chapter the well-known graph products of k-dimensional unit-distance graphs are studied; the chapter summarizes the results from [58]. The third chapter disproves the wrong assumption that Heawood graph is not a unit-distance graph, by providing the unit-distance coordinatization of it. In the fourth chapter all degenerate unit-distance representations of the Petersen graph in the Euclidean plane are presented and some relationships among them are observed; see [57]. In the following chapter generalized Petersen graphs and I-graphs are observed. Necessary and sufficient conditions for two I-graphs to be isomorphic are given. As a corollary it is shown that a large subclass of I-graphs can be drawn with unit-distances in the Euclidean plane by using the representation with a rotational symmetry. Conjectures concerning unit-distance coordinatizations and highly-degenerate unit-distance representations of I-graphs are stated and verified for all I-graphs up to 2000 vertices. In the sixth chapter the decision problems that ask about the existence of a degenerate k-dimensional unit-distance representation or coordinatization of a given graph are shown to be NP-complete. In the last chapter of the thesis a heuristics that draws a given graph in the Euclidean plane by minimizing the quotient of the longest and the shortest edge length is presented; see SPE algorithm in [1]. The dilation coefficient of a graph is introduced and theoretically obtained bounds for the dilation coefficient of a complete graph are given. The calculated upper bounds for the dilation coefficients of complete graphs are compared to the values obtained by three graph-drawing algorithms, see [62].

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