Editing Outerplanar graph (section)

{{Short description|Non-crossing graph with vertices on outer face}}
[[File:Triangulation 3-coloring.svg|thumb|A maximal outerplanar graph and its 3-coloring]]
[[File:Finite-3-regular-graph-4-vertices.png|thumb|The [[complete graph]] K<sub>4</sub> is the smallest planar graph that is not outerplanar.]]
In [[graph theory]], an '''outerplanar graph''' is a graph that has a [[planar graph|planar drawing]] for which all vertices belong to the outer face of the drawing.

Outerplanar graphs may be characterized (analogously to [[Wagner's theorem]] for planar graphs) by the two [[forbidden minor]]s {{math|''K''<sub>4</sub>}} and {{math|''K''<sub>2,3</sub>}}, or by their [[Colin de Verdière graph invariant]]s.
They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has [[Degeneracy (graph theory)|degeneracy]] and [[treewidth]] at most&nbsp;2.

The outerplanar graphs are a subset of the [[planar graph]]s, the subgraphs of [[series–parallel graph]]s, and the [[circle graph]]s. The '''maximal outerplanar graphs''', those to which no more edges can be added while preserving outerplanarity, are also [[chordal graph]]s and [[visibility graph]]s.