Nested hierarchies in planar graphs

TL;DR

This paper introduces a hierarchical framework for maximal planar graphs by decomposing them into bubble subgraphs connected in a tree structure, revealing community-like subdivisions and their relations.

Contribution

It constructs a novel partial order on 3-cliques that defines a unique hierarchy and decomposes the graph into bubbles linked in a tree structure.

Findings

Maximal planar graphs can be uniquely decomposed into bubbles.

Bubbles form communities within the graph.

The hierarchy is represented as a tree connecting bubbles.

Abstract

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G 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 G into communities and the tree structure defines the hierarchical relations between these communities.