Recall a graph is *chordal* if it contains no induced cycle of length 4 or more, and *outerplanar* if it has a crossing-free embedding in the plane such that all vertices are on the same face. While studying a certain problem, I came across *chordal outerplanar graphs*, which are graphs that are both chordal and outerplanar. [ISGCI](http://www.graphclasses.org/) didn't know of such a class. In fact, the closest thing I was able to find was the class of [chordal $\cap$ planar](http://www.graphclasses.org/classes/gc_985.html) graphs.



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> Have chordal outerplanar graphs been studied before somewhere? What kind of properties do they have? Are they perhaps equivalent to some well-known (or a better known) graph class?
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For example, it seems to me that every inner face of such a graph is a triangle. The motivation for the third question is that I got at most 5 hits when googling for such graphs, so I wonder if such graphs might be commonly known as something other than chordal outerplanar graphs.



Have chordal outerplanar graphs been studied before?
Yes, every inner face must be a triangle, as chords drawn outside a cycle prevent a drawing from being outerplanar. Drawings of chordal graphs are then just triangulations of polygons; conversely, every triangulated polygon is outerplanar (certainly) and chordal (by induction, as cycles are sub-[triangulated polygons]).


