<p>The <a href="https://en.wikipedia.org/wiki/Median_graph" rel="nofollow">median graphs</a> are a subclass of modular graphs with almost the same definition but where the vertex w is always unique. The <a href="https://en.wikipedia.org/wiki/Squaregraph" rel="nofollow">squaregraphs</a> (plane graphs in which all bounded faces are quadrilaterals and all vertices either have degree 4 or belong to the unbounded face) form a natural and interesting subclass of the planar median graphs. I believe that all planar median graphs can be constructed by gluing together cubes and squaregraphs, but I don't think this result has been published.</p>

CC BY-SA 3.0
2016-09-07T07:19:25.950Z
I posted another partial answer (the graphs of 2d modular lattices) on my blog at http://11011110.livejournal.com/334654.html — but I think this still is far from a more general classification.
David Eppstein
CC BY-SA 3.0
2016-09-03T11:03:13.853Z
Thank you for your reply! I am hoping for a more general classification or subclass of planar modular or pseudomodular graphs; the existence of a unique median, in particular, is a little too strong of a condition
aellab
CC BY-SA 3.0
2016-09-02T05:17:50.473Z
2016-09-02T05:17:50.473Z
David Eppstein
<p>We say that a graph G is modular if for every three vertices x,y,z there exists a vertex w that lies on a shortest path between every two of x, y, z. Pseudo-modular (or "3-Helly") graphs are defined in the following link: <a href="http://www.graphclasses.org/classes/gc_203.html" rel="nofollow">http://www.graphclasses.org/classes/gc_203.html</a>. </p>

<p>I am wondering if there are interesting examples of planar graphs which are modular or pseudo-modular, or an interesting characterization of planar graphs which have these properties.</p>

<graph-theory><planar-graphs><metric-spaces>
Subclasses or characterizations of modular or pseudo-modular planar graphs
aellab
# Subclasses or characterizations of modular or pseudo-modular planar graphs

(Asked by: username_0 on 2016-09-02)

We say that a graph G is modular if for every three vertices x,y,z there exists a vertex w that lies on a shortest path between every two of x, y, z. Pseudo-modular (or "3-Helly") graphs are defined in the following link: <http://www.graphclasses.org/classes/gc_203.html>.

I am wondering if there are interesting examples of planar graphs which are modular or pseudo-modular, or an interesting characterization of planar graphs which have these properties.

**Tags:** graph-theory, planar-graphs, metric-spaces


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## 1 answers

### Answer by username_1 on 2016-09-02. Score: 2

The [median graphs](https://en.wikipedia.org/wiki/Median_graph) are a subclass of modular graphs with almost the same definition but where the vertex w is always unique. The [squaregraphs](https://en.wikipedia.org/wiki/Squaregraph) (plane graphs in which all bounded faces are quadrilaterals and all vertices either have degree 4 or belong to the unbounded face) form a natural and interesting subclass of the planar median graphs. I believe that all planar median graphs can be constructed by gluing together cubes and squaregraphs, but I don't think this result has been published.

> **Comments:**
> **username_0**: Thank you for your reply! I am hoping for a more general classification or subclass of planar modular or pseudomodular graphs; the existence of a unique median, in particular, is a little too strong of a condition  
> **username_1**: I posted another partial answer (the graphs of 2d modular lattices) on my blog at http://11011110.livejournal.com/334654.html — but I think this still is far from a more general classification.  

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