Patchworks and metablocks enumeration

Abstract

A cutpoint of a graph is a point whose removal increases the number of connected components. A block graph is connected, not reduced to a point and has no cutpoint. A cutedge is the association of an edge and his two adjacent vertices such that their removal increase the number of components. The graphs considered here specify the notion of blocks. We call metablock a connected graph with no cutpoint and no cutedge. A graphical patchwork is a graph without a cutpoint but with at least one cutedge. We show here how to enumerate labeled metablocks and patchworks by obtaining their exponential generating functions according to their number of edges and vertices. This problem is quite interesting because of its originality. A particular consequence is the observation that almost all graphs are patchworks.