Almost All Even Yao-Yao Graphs Are Spanners

It is an open problem whether Yao-Yao graphs YY_{k} (also known as sparse-Yao graphs) are all spanners when the integer parameter k is large enough. In this paper we show that, for any integer k >= 42, the Yao-Yao graph YY_{2k} is a t_k-spanner, with stretch factor t_k = 6.03+O(k^{-1}) when k tends to infinity. Our result generalizes the best known result which asserts that all YY_{6k} are spanners for k >= 6 [Bauer and Damian, SODA\u2713]. Our proof is also somewhat simpler

Odd Yao-Yao Graphs are Not Spanners

It is a long standing open problem whether Yao-Yao graphs YY_{k} are all spanners [Li et al. 2002]. Bauer and Damian [Bauer and Damian, 2012] showed that all YY_{6k} for k >= 6 are spanners. Li and Zhan [Li and Zhan, 2016] generalized their result and proved that all even Yao-Yao graphs YY_{2k} are spanners (for k >= 42). However, their technique cannot be extended to odd Yao-Yao graphs, and whether they are spanners are still elusive. In this paper, we show that, surprisingly, for any integer k >= 1, there exist odd Yao-Yao graph YY_{2k+1} instances, which are not spanners