Nested hierarchies in planar graphs

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph GG and defines a unique hierarchy. We demonstrate that GG is the union of a set of special subgraphs, named ‘bubbles’, that are themselves maximal planar graphs. The graph GG is retrieved by connecting these bubbles in a tree structure where neighboring bubbles are joined together by a 3-clique. Bubbles naturally provide the subdivision of GG into communities and the tree structure defines the hierarchical relations between these communities.

► We construct unique hierarchies on a maximal planar graph.
► Separating property of cycles defines a partial order on the set of 3-cliques.
► The poset defines a complete set of special subgraphs called ‘bubbles’.
► The bubbles and separating 3-cliques constitute a unique bubble hierarchical tree.
► Communities and hierarchy naturally emerge from the bubble hierarchy.