  The existing combinatorial methods for iso-surface computation are efficient
for pure visualization purposes, but it is known that the resulting
iso-surfaces can have holes, and topological problems like missing or wrong
connectivity can appear. To avoid such problems, we introduce a
graph-theoretical method for the computation of iso-surfaces on cuboid meshes
in $\mathbb{R}^3$. The method for the generation of iso-surfaces employs
labeled cuboid graphs $G(V,E,\mathcal{F})$ such that $V$ is the set of vertices
of a cuboid $C\subset\mathbb{R}^3$, $E$ is the set of edges of $C$ and
$\mathcal{F}\,:\,V\rightarrow [0,1]$. The nodes of $G$ are weighted by the
values of $\mathcal{F}$ which represents the volumetric information, e.g.\ from
a Volume of Fluid method. Using a given iso-level $c\in (0,1)$, we first obtain
all iso-points, i.e.\ points where the value $c$ is attained by the
edge-interpolated $\mathcal{F}$-field. The iso-surface is then built from
iso-elements which are composed of triangles and are such that their polygonal
boundary has only iso-points as vertices. All vertices lie on the faces of a
single mesh cell.
  We give a proof that the generated iso-surface is connected up to the
boundary of the domain and it can be decomposed into different oriented
components. Two different components may have discrete points or line segments
in common. The graph-theoretical method for the computation of iso-surfaces
developed in this paper enables to recover local information of the iso-surface
that can be used e.g.\ to compute discrete mean curvature and to solve surface
PDEs. Concerning the computational effort, the resulting algorithm is as
efficient as existing combinatorial methods.
