A plane graph $G$ defines a cyclic ordering $O(v) = \langle v_1, v_2, \dotsc, n_{\deg(v)}\rangle$ on the neighborhood $N(v)$ of each vertex $v \in V(G)$. A non-intersecting Eulerian circuit $C$ is an Eulerian circuit such that for every vertex $v \in V(G)$ and every choice of four edges $\{a,v\}, \{b,v\}, \{c,v\}, \{d,v\} \in E(G)$ containing $v$, if the cyclic ordering $O(v)$ restricted to $\{a,b,c,d\}$ is $\langle a,b,c,d \rangle$ and $\{a,c\} \in C$, then $\{b,d\} \not\in C$.
I have found several references that claim or prove that every plane Eulerian graph contains a non-intersecting Eulerian circuit and at least one reference that shows how to find one in polynomial time.
Is there a standard reference (preferably a book on graph theory) that shows how to compute a non-intersecting Eulerian circuit in a plane Eulerian graph in polynomial time?
