The halfedge description $M=(E,s,\rho)$ of an oriented surface contains all the information. It is nevertheless convenient to introduce some more derived stucture. For an edge $e\in E$ we define
$\textrm{left}(e)=$ the face (cycle of $s$) containing $e$
$\textrm{right}(e)=$ the face (cycle of $s$) containing $\rho(e)$
$\textrm{start}(e)=$ the vertex (cycle of $s\circ\rho$) containing $e$
$\textrm{end}(e)=$ the vertex (cycle of $s\circ\rho$) containing $\rho(e)$
$\textrm{unorient}(e)=$ the unoriented edge (cycle of $\rho$) containing $e$.
Denote by $F$ the set of all faces of $M$, by $V$ the set of all vertices and by $\tilde{E}$ the set of all unoriented edges.
If we look at the quintuple $(E,V,\rho,\textrm{start},\textrm{end})$ we see finite sets $E,V$, an involution $\rho: E \rightarrow E$ without fixed points and two maps $\textrm{start},\textrm{end}: E \rightarrow V$ such that
$\textrm{start} \circ \rho = \textrm{end}$
This is a one of the possible descriptions of a general finite graph. This graph is called the 1-skeleton or edge-graph of $M$. In a computer graphics context one might also call it the wireframe of $M$.