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  1. Does graph shown below from the paper Dissection Graphs of Planar Point Sets by P. Erdos, L. Lovasz, A. Simmons, and E.G. Straus have a name?

  2. Does it come from a family of related graphs?

This comes from the Erdos, Lovasz Paper

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This graph is a chordal graph (triangulated graphs) – M.M Jul 25 '15 at 16:50
up vote 0 down vote accepted

As stated in the comments, this is a chordal graph. It's also a block graph with blocks $B_1=\{p'_1,p''_1\}$, $B_2=\{p'_3,p''_3\}$, $B_3=\{p'_5,p''_5\}$, and $B_4=\{p''_1,p'_2,p''_2,p''_3,p'_4,p''_4,p''_5,p'_6,p''_6\}$.

Consider the cycle $C=\{p'_6,p''_6,p''_4,p''_2\}$. The graph is not strongly chordal ($C$ doesn't contain an odd chord). The graph is not a split graph ($C$ is a $4$-cycle, which is a forbidden subgraph for split graphs).

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I don't think this is a block graph though. In a block graph, the blocks should be complete. You claim $B_4$ is a block, but it is not a clique. – mrm Oct 14 '15 at 13:54

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