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Suppose given a $1$-separated net $\Gamma\subset\mathbb R^2$. Is it true or false that there exists $\delta>0$ and a $\delta$-isoradial graph containing $\Gamma$ as a subset of its vertices?

(I am also interested in partial results in this direction.)

edit: a $\delta$-isoradial graph in $\mathbb R^2$ is a graph each of whose faces is a finite polygon inscribed in some circle of radius $\delta$.

edit2: A case where the answer is "yes": if we have a graph whose faces are $\delta$-rhombi then by considering only odd vertices gives us an isoradial graph. It will help to consider such graphs instead of $\delta$-isoradial ones. If $\Gamma$ is included in the graph of a continuous function $G_f:=\{(x,f(x)):x\in\mathbb R\}$ then we may approximate $G_f$ by a polygonal $\delta$-path $P$ which comes from a graph of a function $\tilde f$ and contains $\Gamma$ as a subset of its odd vertices. Now extend $P$ to a path of $\delta$-rhombi and then extend to a $\delta$-rhombic tiling of $\mathbb R^2$ e.g. by periodicity.

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better give a def of isoradial graph. – Anton Petrunin Apr 24 '13 at 22:09

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