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Catafusenes (catacondensed simply connected polyhexes) annelated to a core are studied. Schemes for handling the enumeration of these systems with four appendages or less are given. In this survey the formalism of combinatorial summations is employed, but the exposition is supported by applications of generating functions. Special cases of interest occur when the core is a g-gon (g-membered polygon). Then the systems are referred to as catacondensed mono-g-fusenes, which are subclasses of the important mono-g-polyhexes.

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