Classical mereology is based on the assumption that any two underlapping elements have a sum, yet there are many domains (such as manufacturing assemblies, molecular structure, gene sequences, and convex time intervals) in which this assumption is not valid. In such domains, mereological sums must be connected objects. However, there has been little work in providing an axiomatization of such a mereology. Based on the observation that the underlying structures in these domains are represented by graphs, we propose a new mereotopology that axiomatizes the connected induced subgraph containment ordering for a graph, and then identify an axiomatization of the mereology that is a module of the mereotopology.