Do you know anything special about that kind of planar graphs? An article that covers these graphs might be helpful.
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$\begingroup$ Why this particular class of graphs? $\endgroup$– Noah SchweberCommented Dec 10, 2015 at 23:56
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1$\begingroup$ If you add the condition that the graph is self-dual, then you can say some things, but without more conditions I doubt you can say much. $\endgroup$– Douglas ZareCommented Dec 10, 2015 at 23:59
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$\begingroup$ I came across to those graphs when I was trying to solve a problem. I need some properties to approach that problem. $\endgroup$– TranscendentalCommented Dec 10, 2015 at 23:59
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2$\begingroup$ It sounds like you should try another approach, then. $\endgroup$– Douglas ZareCommented Dec 11, 2015 at 0:00
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1 Answer
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Unlike some other planar graphs, these ones always contain at least one triangular face and at least one vertex of degree $\le 3$. The reason for the first property is that, by Euler's formula, the number of edges in a planar graph with no triangles is at most $2n-4$, but your graphs always have exactly $2n-2$ edges. The second property is just the dual of the first.
As Douglas Zare already said, I doubt they have any more interesting properties than that, though. They can be made by mixing parts of graphs that are dense and other parts that are sparse until you have the right overall balance of edges and faces, so near any particular vertex they can look like any other planar graph.
answered Dec 11, 2015 at 3:40