Block-cutvertex trees and block-cutvertex partitions Original Research Article

The block-cutvertex graph of the connected graph G, denoted bc(G), is the graph whose vertices are the blocks and cutvertices of G. The edges of bc(G) join cutvertices with those blocks to which they belong. Gallai, Harary and Prins defined this concept and showed that a graph G is the block-cutvertex graph of some connected graph H if and only if G is a tree in which the distance between any two leaves is even. A block-cutvertex partition of the tree T is a collection {T1,…,Tk} of block-cutvertex trees such that each Ti is a subtree of T and each edge of T is in exactly one Ti. We prove that a tree has a block-cutvertex partition if and only if it does not have a perfect matching. Various concepts and algorithms related to block-cutvertex partitions will be presented.