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Concept
Apollonian network
Summary
In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
An Apollonian network may be formed, starting from a single triangle embedded in the Euclidean plane, by repeatedly selecting a triangular face of the embedding, adding a new vertex inside the face, and connecting the new vertex to each vertex of the face containing it. In this way, the triangle containing the new vertex is subdivided into three smaller triangles, which may in turn be subdivided in the same way.
The complete graphs on three and four vertices, K3 and K4, are both Apollonian networks. K3 is formed by starting with a triangle and not performing any subdivisions, while K4 is formed by making a single subdivision before stopping.
The Goldner–Harary graph is an Apollonian network that forms the smallest non-Hamiltonian maximal planar graph. Another more complicated Apollonian network was used by to provide an example of a 1-tough non-Hamiltonian maximal planar graph.
As well as being defined by recursive subdivision of triangles, the Apollonian networks have several other equivalent mathematical characterizations. They are the chordal maximal planar graphs, the chordal polyhedral graphs, and the planar 3-trees. They are the uniquely 4-colorable planar graphs, and the planar graphs with a unique Schnyder wood decomposition into three trees. They are the maximal planar graphs with treewidth three, a class of graphs that can be characterized by their forbidden minors or by their reducibility under Y-Δ transforms. They are the maximal planar graphs with degeneracy three.
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