In the (very nice) article of Goldstein and Turner untitled *Applications of Topological Graph Theory to Group Theory*, the following definitions can be found:

**Definitions:** A *circle graph* is a pair $(G,S)$ where $G$ is a trivalent graph and $S$ a Hamiltonian cycle. (Topologically, a circle graph consists of a circle together with a finite number of arcs whose end points are disjoint on the circle.)

An *embedding* of a circle graph is defined as a piecewise linear embedding $f : (G,S) \to (M,\partial M)$ where $M$ is obtained from a closed 2-manifold $\hat{M}$ by removing an open 2-cell.

Finally, the *oriented genus* $\gamma^0(G,S)$ of $(G,S)$ is the genus of the oriented manifold $\hat{M}$ of minimal genus so that there exists an embedding of $(G,S)$ into $(M, \partial M)$

If $g(G,S)$ denotes the usual genus of the associated graph, the inequality $g(G,S) \leq \gamma^0(G,S)$ is clear. Does the equality hold?