I think Artem is on the right track with perfection: As cographs are $P_4$-free, cograph+v is $C_5$-free (and $C_{2k+1}$-free and $\overline{C}_{2k+1}$-free, $k>1$) and so they are perfect graphs. This means the only thing that is pushing the chromatic number up is clique size. So if $\chi(G+v) = \chi(G) + 1$, it is because v has increased the maximum ...


If I would like to know something about intersection models, the first reference I would check is the "Topics in Intersection Graph Theory" by McKee and McMorris. Theorem 1.5 answers your (combinatorial) question.

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