The compact symmetric multicategories (CSMs) introduced by Joyal and Kock in their 2011 note 'Feynman Graphs, and Nerve Theorem for Compact Symmetric Multicategories' [JK11] directly generalise a number of unital operad types, such as wheeled properads, that admit a contraction operation as well as an operadic multiplication. These structures are known to exhibit strange behaviour related to the contraction of units, and this is problematic for [JK11]. In this thesis, I modify the construction of [JK11] to obtain non unital (coloured) modular operads as algebras for a monad defined in terms of connected graphs, and use this as a foundation for a new construction of CSMs based on special graph morphisms. A corresponding nerve theorem characterises CSMs in terms of a Segal condition. This construction sheds light, and provides some control, on the behaviour of the contracted units.