A block graph, also called a clique tree, is a simple graph in which every block is a complete graph. The numbers of connected block graphs on , 2, ... vertices are 1, 1, 2, 4, 9, 22, 59, 165, 496, ... (OEIS A035053).

Block graphs are chordal, distance-hereditary, geodetic, perfect, and Ptolemaic.

Classes of graphs which are block graphs include windmall graphs and trees.

A graph is a block graph if the lagest two of the distances

are equal.

More things to try:

Weisstein, Eric W. "Block Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BlockGraph.html