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I am interested in the properties of the following subclass of split graphs:

The class consists of all split graphs $G=(C\cup I)$ where $C$ is a clique and $I$ an independent set, and every pair of vertices in $I$ have at least one common neighbor in $C$.

Does this class of graphs have a special name? Has this class and its properties been studied? If so, what would be some good references for this?

flag	asked Aug 24 2010 at 11:47 gphilip 75●3

Jukka Suomela · Accepted Answer · 2010-08-24 21:10:04Z UTC

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Split graphs of diameter 2?

answered Aug 24 2010 at 21:10

Jukka Suomela
281●1●6

I guess that's all that can be said about these. I searched far and wide, these don't seem to have any specific name. Thank you. – gphilip Aug 25 2010 at 3:42

Vinicius dos Santos · Answer 2 · 2010-09-02 03:35:28Z UTC

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I don't know if this can help you, but it's easy to see that your class is a superclass of connected threshold graphs. I think that the inclusion is strict, since $S_3$(link) is in your class, but is not a threshold graph.

answered Sep 2 2010 at 3:35

Vinicius dos Santos
41●2

Thank you, that helps a bit. – gphilip Sep 19 2010 at 23:48

Have this subclass of split graphs been studied before?

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