Geometric graphs are frequently used to model a wireless ad hoc network in order to build efficient routing algorithms. The network topology in mobile wireless networks may often change therefore position-based routing that uses the idea
of localized routing has an advantage over other types of routing protocols. Since a wireless network has limited memory and energy resources, topology control has an important role in enhancing certain desirable properties of
these networks. To achieve the goal of topology control, spanning subgraphs of the UDG graph such as Relative Neighborhood Graph, Gabriel Graph and Yao Graph are constructed, which are then routed upon rather than the original UDG.

In this thesis we introduce a spanning subgraph of the UDG graph that is a variation of the Displaced Apex Adaptive Yao (DAAY) graph. In this subgraph an exclusion zone based on a parabola is defined with respect to each non-excluded
nearest neighbor and positioned on each node, instead of a cone as with the DAAY subgraph, such that each nearest neighbor is inside the parabola. It also lets the apex of the parabola to move along the line segment between the node and its neighbor. The subgraph has two adjustable parameters, one each for the position of the apex with respect to the nearest neighbor, and the width of the parabola. Thus a directed or undirected spanning subgraph
of a UDG is constructed. We show that this spanning subgraph is connected, has a conditional bounded out-degree, is a t-spanner with bounded stretch factor, and contains the Euclidean minimum spanning tree as a subgraph.
Experimental comparisons with related spanning subgraphs are also presented.