Enumerating Consecutive and Nested Partitions for Graphs

Consecutive & nested partitions have been extensively studied in the set-partition problem as tools
with which to search efficiently for an optimal partition. We extend the study of consecutive and
nested partitions on a set of integers to the vertex-set of a graph. A subset of vertices is considered
consecutive if the subgraph induced by the subset is connected. In this sense the partition problem on
a set of integers can be treated as a special case when the graph is a line. In this paper we give the
number of consecutive & nested partitions when the graph is a cycle. We also give a partial order
on general graphs with respect to these numbers.