Editing Outerplanar graph (section)

==Related families of graphs==
[[File:Cactus graph.svg|thumb|A [[cactus graph]]. The cacti form a subclass of the outerplanar graphs.]]
Every outerplanar graph is a [[planar graph]]. Every outerplanar graph is also a subgraph of a [[series–parallel graph]].<ref>{{harvtxt|Brandstädt|Le|Spinrad|1999}}, p. 174.</ref> However, not all planar series–parallel graphs are outerplanar. The [[complete bipartite graph]] ''K''<sub>2,3</sub> is planar and series–parallel but not outerplanar. On the other hand, the [[complete graph]] ''K''<sub>4</sub> is planar but neither series–parallel nor outerplanar. Every [[tree (graph theory)|forest]] and every [[cactus graph]] are outerplanar.<ref>{{harvtxt|Brandstädt|Le|Spinrad|1999}}, p. 169.</ref>

The [[planar dual|weak planar dual]] graph of an embedded outerplanar graph (the graph that has a vertex for every bounded face of the embedding, and an edge for every pair of adjacent bounded faces) is a forest, and the weak planar dual of a [[Halin graph]] is an outerplanar graph. A planar graph is outerplanar if and only if its weak dual is a forest, and it is Halin if and only if its weak dual is biconnected and outerplanar.<ref>{{harvtxt|Sysło|Proskurowski|1983}}.</ref>

There is a notion of degree of outerplanarity.  A 1-outerplanar embedding of a graph is the same as an outerplanar embedding.  For ''k''&nbsp;&gt;&nbsp;1 a planar embedding is said to be [[K-outerplanar graph|''k''-outerplanar]] if removing the vertices on the outer face results in a (''k''&nbsp;&minus;&nbsp;1)-outerplanar embedding.  
A graph is ''k''-outerplanar if it has a ''k''-outerplanar embedding.<ref>{{harvtxt|Kane|Basu|1976}}; {{harvtxt|Sysło|1979}}.</ref>

An [[1-planar graph#Generalizations and related concepts|outer-1-planar graph]], analogously to [[1-planar graph]]s can be drawn in a disk, with the vertices on the boundary of the disk, and with at most one crossing per edge.

Every maximal outerplanar graph is a [[chordal graph]]. Every maximal outerplanar graph is the [[visibility graph]] of a [[simple polygon]].<ref>{{harvtxt|El-Gindy|1985}}; {{harvtxt|Lin|Skiena|1995}}; {{harvtxt|Brandstädt|Le|Spinrad|1999}}, Theorem 4.10.3, p. 65.</ref> Maximal outerplanar graphs are also formed as the graphs of [[polygon triangulation]]s. They are examples of [[k-tree|2-trees]], of [[series–parallel graph]]s, and of [[chordal graph]]s.

Every outerplanar graph is a [[circle graph]], the [[intersection graph]] of a set of chords of a circle.<ref>{{harvtxt|Wessel|Pöschel|1985}}; {{harvtxt|Unger|1988}}.</ref>