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Genus Ranges of 4-regular Rigid Vertex Graphs and Their Chord Diagrams
by
Natasha Jonoska
University of South Florida
Coauthors: Masahico Saito
A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. Four-regular rigid vertex graphs are closely related to chord diagrams (a circle with line segments, called chords, whose endpoints are attached to distinct points on the circle). These structures appear in studies of DNA recombination processes. We present results about the orientable genus ranges of these graphs and the chord diagrams. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs, and for the chord diagrams, are sets of consecutive integers. We address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. We present the intervals that can be realized as genus ranges for graphs with fixed number of vertices (chord diagrams with fixed number of chords) and we provide constructions of graphs that realize these ranges.
Date received: December 29, 2014
Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbjz-76.