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<p><b>Abstract</b>—In wireless ad hoc networks, constructing and maintaining a topology with lower node degrees is usually intended to mitigate excessive traffic load on wireless nodes. However, keeping lower node degrees often prevents nodes from choosing better routes that consume less energy. Therefore, the trade-off is between the node degree and the energy efficiency. In this paper, an adjustable structure, named the <tmath>r{\hbox{-}}{\rm neighborhood}</tmath> graph, is proposed to control the topology. This structure has the flexibility to be adjusted between the two objectives through a parameter <tmath>r</tmath>, <tmath>0 \le r \le 1</tmath>. More explicitly, for any set of <tmath>n</tmath> nodes, the maximum node degree and power stretch factor can be bounded from above by some decreasing and increasing functions of <tmath>r</tmath>, respectively. Specifically, the bounds can be constants in some ranges of <tmath>r</tmath>. Even more, the <tmath>r{\hbox{-}}{\rm neighborhood}</tmath> graph is a general structure of both <tmath>RNG</tmath> and <tmath>GG</tmath>, two well-known structures in topology control. Compared with <tmath>YG_{k}</tmath>, another famous adjustable structure, our method always results in a connected planar with symmetric edges. To construct this structure, we investigate a localized algorithm, named <it>PLA</it>, which consumes less transmitting power during construction and executes efficiently in <tmath>O(n\log n)</tmath> time.</p>
Wireless ad hoc networks, topology control, energy-efficient, localized algorithm.

A. A. Jeng and R. Jan, "The r-Neighborhood Graph: An Adjustable Structure for Topology Control in Wireless Ad Hoc Networks," in IEEE Transactions on Parallel & Distributed Systems, vol. 18, no. , pp. 536-549, 2007.
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