Block graph

[6] Because of this property, in a connected block graph, every set of vertices has a unique minimal connected superset, its closure in the convex geometry.

[1] Block graphs are chordal, distance-hereditary, and geodetic.

[7] Block graphs are examples of pseudo-median graphs: for every three vertices, either there exists a unique vertex that belongs to shortest paths between all three vertices, or there exists a unique triangle whose edges lie on these three shortest paths.

Line graphs of trees have been used to find graphs with a given number of edges and vertices in which the largest induced subgraph that is a tree is as small as possible.

The largest triangular cactus in any graph may be found in polynomial time using an algorithm for the matroid parity problem.