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When we study forbidden subgraph conditions guaranteeing graphs to have someproperties, a claw (or $K_{1,3}$) frequently appears as one of forbiddensubgraphs. Recently, Furuya and Tsuchiya compared two classes generated bydifferent forbidden pairs containing a claw, and characterized one of suchclasses. In this paper, we give such characterization for three new classes.Furthermore, we give applications of our characterizations to some forbiddensubgraph problems.

Difference of forbidden pairs containing a claw