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Star-cutsets and perfect graphs. (English) Zbl 0674.05058
This paper presents a general structure for understanding various known techniques for producing a new perfect graph $$G^*$$ out of a given pair of old perfect graphs $$G_ 1,G_ 2$$. A common property of these techniques is that: if an induced subgraph F of $$G^*$$ (with F having at least three vertices) is an induced subgraph of neither $$G_ 1$$ nor $$G_ 2$$, then F has a star-cutset or its complement $$F^ c$$ is disconnected. Here a star-cutset is a set of vertices that form a cutset and whose induced subgraph contains some vertex adjacent to all others in the cutset. The star closure $${\mathcal C}^*$$ of a class $${\mathcal C}$$ of graphs is recursively defined by the rules: (i) if $$G\in {\mathcal C}$$, then $$G\in {\mathcal C}^*$$; and (ii) if G or $$G^ c$$ has a star-cutset and if G-v$$\in {\mathcal C}^*$$ for all v in G, then $$G\in {\mathcal C}^*$$. The author proves that two classes of strongly perfect graphs, the Meyniel graphs and perfectly orderable graphs, are in the star closure of the class of bipartite graphs.

##### MSC:
 05C99 Graph theory 05C35 Extremal problems in graph theory
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##### References:
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