Computing the Number and Average Size of Connected Sets in Planar 3-Trees

A vertex set in a graph is a connected set if it induces a connected subgraph. For a tree T, each subgraph induced by a connected set of T is actually a subtree of T. The number and average size of subtrees of a tree T are two well-studied parameters. Yan and Yeh developed a linear-time algorithm for computing the number of subtrees in a tree through “generating function”. In this paper, we present linear-time algorithms for computing the number and average size of connected sets in a planar 3-tree.