[Paper Review] New representation results for planar graphs

Farhad Shahrokhi|arXiv (Cornell University)|Feb 22, 2015

Advanced Graph Theory Research3 references12 citations

TL;DR

This paper presents a new universal representation theorem showing that any planar graph can be expressed as the intersection of one chordal graph, four co-bipartite graphs, and one unit interval graph. The key innovation lies in introducing clique cover width as a generalization of bandwidth, enabling a bounded representation for planar graphs and extending to graphs on surfaces and those excluding a minor with crossing number at most one.

ABSTRACT

A universal representation theorem is derived that shows any graph is the intersection graph of one chordal graph, a number of co-bipartite graphs, and one unit interval graph. Central to the the result is the notion of the clique cover width which is a generalization of the bandwidth parameter. Specifically, we show that any planar graph is the intersection graph of one chordal graph, four co-bipartite graphs, and one unit interval graph. Equivalently, any planar graph is the intersection graph of a chordal graph and a graph that has {clique cover width} of at most seven. We further describe the extensions of the results to graphs drawn on surfaces and graphs excluding a minor of crossing number of at most one.

Motivation & Objective

• To develop a new representation framework for planar graphs using intersection graphs of specific graph classes.
• To generalize the bandwidth parameter via clique cover width (CCW) for structural and algorithmic applications.
• To extend the representation results to graphs embedded on surfaces and those excluding a minor with crossing number ≤1.
• To establish connections between graph decomposition, intersection representations, and computational efficiency via separators.
• To provide a polynomial-time construction of intersection models with bounded clique cover width for planar and related graph classes.

Proposed method

• Introduce the clique cover width (CCW) as a generalization of bandwidth, measuring the maximum distance between cliques in a vertex partition that respects edge adjacency.
• Prove a universal representation theorem: any graph is the intersection of a chordal graph and a graph with bounded CCW, based on tree decompositions and vertex partitioning.
• Apply the universal theorem to planar graphs by constructing a tree decomposition with bounded intersection per level, leading to CCW ≤ 7.
• Use Theorem 1.1 to decompose graphs with bounded CCW into ⌈log w⌉+1 co-bipartite graphs and one unit interval graph.
• Extend results to graphs of genus g by induction, showing CCW bounds grow as O(log g).
• Handle minor-excluded graphs with crossing number ≤1 by decomposing into clique sums of planar or low-treewidth graphs and combining representations via clique sum operations.

Experimental results

Research questions

• RQ1 Can planar graphs be represented as the intersection of a bounded number of structured graph classes, such as chordal, co-bipartite, and unit interval graphs?
• RQ2 How does the new clique cover width parameter relate to bandwidth and what structural insights does it provide for graph decomposition?
• RQ3 To what extent can the intersection representation framework be extended to graphs on surfaces or those excluding a minor with small crossing number?
• RQ4 What is the computational complexity of constructing such intersection representations, and can they yield efficient separators?
• RQ5 Can the representation framework be used to derive or generalize known results like the planar separator theorem?

Key findings

• Any planar graph is the intersection of one chordal graph, four co-bipartite graphs, and one unit interval graph, as a direct consequence of the universal representation theorem and Theorem 1.1.
• The clique cover width of the auxiliary graph in the representation is bounded by 7 for planar graphs, derived from a tree decomposition with maximum level intersection size t* ≤ 4.
• The representation extends to graphs of genus g, where the number of co-bipartite graphs needed is O(log g), showing a logarithmic dependence on topological complexity.
• For graphs excluding a minor H with crossing number ≤1, the number of co-bipartite graphs is bounded by O(log C_H), where C_H is a superexponential function of |V(H₀)| and |E(H₀)|.
• The construction is polynomial-time for all cases except the minor-excluded case, and the framework implies the planar separator theorem via small separator construction.
• The clique cover width is shown to be NP-hard to compute, due to its close relationship with the bandwidth problem.