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Hyperbolicity, injective hulls, and Helly graphs

Guarnera, Heather M
http://rave.ohiolink.edu/etdc/view?acc_num=kent1594058509433274

Abstract Details

2020, PHD, Kent State University, College of Arts and Sciences / Department of Computer Science.
The δ-hyperbolicity of a graph can be viewed as a measure of how close a graph is to a tree metrically; the smaller the hyperbolicity of a graph, the closer it is metrically to a tree. For every graph G there exists a unique smallest Helly graph H(G) into which G isometrically embeds. H(G) is called the injective hull of G, and the hyperbolicity of H(G) is equal to the hyperbolicity of G. Moreover, the minimum distance between a vertex in H(G) and a vertex in G is directly proportional to the hyperbolicity of G, which is a small constant in many real-world networks. Motivated by this, we identify facility location and network analysis problems that are translatable to problems in a graph’s underlying injective hull. Of particular interest for these problems is the eccentricity e(v) of a vertex v, defined as the length of the longest shortest path from v to any other vertex. We show that the injective hull H(G) of a graph G has many nice structural and metric properties that govern the hyperbolicity of G. We analyze the eccentricity function in distance-hereditary graphs (they are 1-hyperbolic) and fully characterize their centers (the set of vertices with minimum eccentricity). We describe the eccentricity terrain of a graph, that is, a topographic perspective of a graph with respect to vertex eccentricities, akin to geomorphometry. We identify terrain shapes such as hills, valleys, and plains, and discover that the length of some shapes depends only on the hyperbolicity. Additionally, we introduce a new graph parameter called the Helly-gap, a measure of a graph’s metric deviation from its injective hull, and show that many well-known graph classes have a small Helly-gap. We present a new linear time algorithm for computing all vertex eccentricities in distance-hereditary graphs. For general graphs, we present various eccentricity approximation algorithms differing in quality in run time, parameterized by the δ-hyperbolicity or in part by the Helly-gap. We then identify several graph classes that are closed under Hellification, that is, G ∈ C implies H(G) ∈ C. We show that the injective hull of several graphs from some well-known classes of graphs are impossible to compute in subexponential time and provide a polynomial time algorithm to construct the injective hull of any distance-hereditary graph.
Feodor Dragan, Ph.D. (Advisor)
Ruoming Jin, Ph.D. (Committee Member)
Gokarna Sharma, Ph.D. (Committee Member)
Artem Zvavitch, Ph.D. (Committee Member)
Jing Li, Ph.D. (Committee Member)
123 p.
Computer Science
injective hull; Gromov hyperbolicity; Helly graphs; vertex eccentricities

Recommended Citations

Citations

  • Guarnera, H. M. (2020). Hyperbolicity, injective hulls, and Helly graphs [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1594058509433274

    APA Style (7th edition)

  • Guarnera, Heather. Hyperbolicity, injective hulls, and Helly graphs. 2020. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1594058509433274.

    MLA Style (8th edition)

  • Guarnera, Heather. "Hyperbolicity, injective hulls, and Helly graphs." Doctoral dissertation, Kent State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=kent1594058509433274

    Chicago Manual of Style (17th edition)

kent1594058509433274
361
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This open access ETD is published by Kent State University and OhioLINK.