In graph theory, a branch of combinatorial mathematics, a block graph or clique tree is a type of undirected graph in which every biconnected component (block) is a clique. Block graphs are sometimes erroneously called Husimi trees (after Kôdi Husimi), but that name more properly refers to cactus graphs, graphs in which every nontrivial biconnected component is a cycle. Block graphs may be characterized as the intersection graphs of the blocks of arbitrary undirected graphs. Characterization Block graphs are exactly the graphs for which, for every four vertices , , , and , the largest two of the three distances , , and are always equal. They also have a forbidden graph characterization as the graphs that do not have the diamond graph or a cycle of four or more vertices as an induced subgraph; that is, they are the diamond-free chordal graphs. They are also the Ptolemaic graphs (chordal distance-hereditary graphs) in which every two nodes at distance two from each other are connected by a unique shortest path, and the chordal graphs in which every two maximal cliques have at most one vertex in common. A graph is a block graph if and only if the intersection of every two connected subsets of vertices of is empty or connected. Therefore, the connected subsets of vertices in a connected block graph form a convex geometry, a property that is not true of any graphs that are not block graphs. 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. The connected block graphs are exactly the graphs in which there is a unique induced path connecting every pair of vertices. Related graph classes Block graphs are chordal, distance-hereditary, and geodetic. The distance-hereditary graphs are the graphs in which every two induced paths between the same two vertices have the same length, a weakening of the characterization of block graphs as having at most one induced path between every two vertices. Because both the chordal graphs and the distance-hereditary graphs are subclasses of the perfect graphs, block graphs are perfect. Every tree, cluster graph, or windmill graph is a block graph. Every block graph has boxicity at most two. 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,