A block graph, also called a clique tree, is a simple graph in which every block is a complete
graph. The numbers of unlabeled connected
block graphs on 
,
2, ... vertices are 1, 1, 2, 4, 9, 22, 59, 165, 496,
... (OEIS A035053).
Block graphs are chordal, distance-hereditary, perfect, and Ptolemaic. Every connected block graph is geodetic.
Classes of graphs which are block graphs include windmill graphs and trees.
A simple graph is a block graph iff, for every four vertices 
, 
,
, and 
in the same connected component,
the largest two of the sums
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
are equal (Howorka 1979), where 
denotes the graph distance.
