Let $G$ be a graph which does not contain a simple cycle $v\_1\ldots v\_k$ and two "crossing" chords $v\_iv\_j$ and $v\_pv\_q$, $i<p<j<q$. An example of such graph is a triangulation of the convex polygon. Is it true that the number of edges in $G$ does not exceed $2n-3$, where $n$ denotes the number of vertices?



Graph in which no cycle has two crossing chords
Thomassen and Toft [[JCTB 31(2):199-224, 1981](https://doi.org/10.1016/S0095-8956(81)80025-1)] showed that any graph with minimum degree at least 3 contains a cycle with two crossing chords from neighbouring vertices on the cycle. The $2n-3$ upper bound follows by induction on $n$, since we may delete a vertex of degree at most $2$.


