The Yao graph for k = 4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners.
O'Rourke, Joseph, "Some Properties of Yao Y4 Subgraphs" (2009). Faculty Publications. Paper 40.