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Here we want to count the number of independent sets of a comparability graph. For Δ(G)=2 , i.e. the maximum degree of a vertex in G is 2, the number of independent sets can be found in linear time. For Δ(G)=4 , the problem is very hard (#P-complete ?). Take for example the grid graph.

I am wondering about the case of Δ(G)=3 . It seems that it is very simple.

Thank you for your advice/comments in advance. :-)