On certain geometric properties of the Yao–Yao graphs

We show that, for any constant $\mathrm{\rho <!--\; rho\; -->}>1$ , there exists an integer constant $k$ such that the Yao–Yao graph with parameter $k$ defined on a civilized unit disk graph is a geometric spanner of stretch factor $\rho$ . This improves the results of Wang and Li in several aspects, as described in the paper. This partially answers an open problem posed by Demaine, Mitchell and O’Rourke about the spanner properties of Yao–Yao graphs. We also show that the Yao–Yao graph with parameter $k=4$ defined on the complete Euclidean graph is not plane.