Spatial Subdivision of Gabriel Graph.

Gabriel graph is one of the well-studied proximity graphs which has a wide range of applications in various research areas such as wireless sensor network, gene flow analysis, geographical variation analysis, facility location, cluster analysis, and so on. In numerous applications, an important query is to find a specific location in a Gabriel graph at where a given set of adjacent vertices can be obtained if a new point is inserted. To efficiently compute the answer of this query, our proposed solution is to subdivide the plane associated with the Gabriel graph into smaller subregions with the property that if a new point is inserted anywhere into a specific subregion then the set of adjacent vertices in the Gabriel graph remains constant for that point, regardless of the exact location inside the subregion. In this paper, we examine these planar subregions, named redundant cells, including some essential properties and sketch an algorithm of running time ({mathcal {O}}(n^2)) to construct the arrangement that yields these redundant cells.