Odd Yao-Yao Graphs may Not be Spanners

It is a long standing open problem whether Yao-Yao graphs $\mathsf{YY}_{k}$ are all spanners. Bauer and Damian \cite{bauer2013infinite} showed that all $\mathsf{YY}_{6k}$ for $k \geq 6$ are spanners. Li and Zhan \cite{li2016almost} generalized their result and proved that all even Yao-Yao graphs $\mathsf{YY}_{2k}$ are spanners (for $k\geq 42$). 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 $k \geq 1$, there exist odd Yao-Yao graph $\mathsf{YY}_{2k+1}$ instances, which are not spanners.

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