There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
Delaunay graphs are almost as good as complete graphs
Discrete & Computational Geometry
Solving the Euclidean bottleneck biconnected edge subgraph problem by 2-relative neighborhood graphs
Discrete Applied Mathematics
The Delauney Triangulation Closely Approximates the Complete Euclidean Graph
WADS '89 Proceedings of the Workshop on Algorithms and Data Structures
20-Relative Neighborhood Graphs Are Hamiltonian
SIGAL '90 Proceedings of the International Symposium on Algorithms
On the Spanning Ratio of Gabriel Graphs and beta-Skeletons
SIAM Journal on Discrete Mathematics
Geometric Spanner Networks
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Wireless ad hoc network can be modeled as a unit disk graph (UDG) in which there is an edge between two nodes if and only if their Euclidean distance is at most one unit. The size of UDG is in O(n2), where n is the number of network nodes. In the literature, the geometric spanners like Relative Neighborhood Graph (RNG), Gabriel Graph (GG), Delaunay Triangulation (Del), Planarized Localized Delaunay Triangulation (PLDel) and Yao Graph are proposed, which are sparse subgraphs of UDG. In this paper, we propose a hierarchy of geometric spanners called the kth order RNG (k-RNG), kth order GG (k-GG), kth order Del (k-Del), and kth order Yao (k-Yao) to reduce the spanning ratio and control topology, sparseness and connectivity. We have simulated these spanners and compared with the existing spanners. The simulation results show that the proposed spanners have better properties in terms of spanning ratio and connectivity by controlling topology and sparseness.