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Spanning Properties of Theta-Theta Graphs

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Combinatorial Optimization and Applications (COCOA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8881))

Abstract

We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of \(k\) cones, for some fixed integer \(k > 1\), and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter \(k = 6k'\) have spanning ratio \(11.67\), for \(k' \ge 6\). In this paper we establish a first spanning ratio of \(7.82\) for Theta-Theta graphs, for the same values of \(k\). We also extend the class of Theta-Theta spanners with parameter \(6k'\), and establish a spanning ratio of \(16.76\) for \(k' \ge 5\). We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to \(4.64\) as \(k'\) increases to \(8\). These are the first results on the spanning properties of Theta-Theta graphs.

This work was supported by NSF grant CCF-1218814.

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Notes

  1. 1.

    The triangular distance from a point \(a\) to a point \(b\) is the side length of the smallest equilateral triangle centered at \(a\) that touches \(b\) and has one horizontal side.

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Authors and Affiliations

  1. Department of Computer Science, Villanova University, Villanova, PA, 19085, USA

    Mirela Damian & Dumitru V. Voicu

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  1. Mirela Damian
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  2. Dumitru V. Voicu
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Correspondence to Mirela Damian .

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Editors and Affiliations

  1. Zhejiang Normal University, Jinhua, Zhejiang, China

    Zhao Zhang

  2. University of Texas, Tyler, Texas, USA

    Lidong Wu

  3. University of Texas, Dallas, Texas, USA

    Wen Xu

  4. University of Texas, Dallas, USA

    Ding-Zhu Du

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Damian, M., Voicu, D.V. (2014). Spanning Properties of Theta-Theta Graphs. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_17

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  • DOI: https://doi.org/10.1007/978-3-319-12691-3_17

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