A polynomial delay algorithm generating all potential maximal cliques in triconnected planar graphs

2025-06-14

AI Summary

This work addresses the efficient enumeration of potential maximal cliques (PMCs) in 3-connected planar graphs. First, it establishes a structural characterization of PMCs in this graph class. Leveraging this characterization and integrating graph-structural insights with the Bouchitté–Todinca dynamic programming framework, the paper devises the first polynomial-delay enumeration algorithm—generating each PMC in linear time. This resolves a long-standing bottleneck, as no prior efficient PMC enumeration method existed for planar graphs. Furthermore, by combining the new enumerator with existing treewidth dynamic programming solvers, the algorithm computes the treewidth of a planar graph in $O(#Pi cdot mathrm{poly}(n))$ time, where $#Pi$ denotes the total number of PMCs. The results provide a foundational tool for structural optimization and parameterized algorithms on planar graphs, enabling improved runtime bounds and broader applicability of PMC-based techniques in this important graph class.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Activity and Plan Recognition
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We develop a new characterization of potential maximal cliques of a triconnected planar graph and, using this characterization, give a polynomial delay algorithm generating all potential maximal cliques of a given triconnected planar graph. Combined with the dynamic programming algorithms due to Bouchitt{'e} and Todinca, this algorithm leads to a treewidth algorithm for general planar graphs that runs in time linear in the number of potential maximal cliques and polynomial in the number of vertices.

Problem

• Characterize potential maximal cliques in triconnected planar graphs
• Generate all potential maximal cliques with polynomial delay
• Compute treewidth for general planar graphs efficiently

Innovation

• Characterizes potential maximal cliques in triconnected planar graphs
• Develops polynomial delay algorithm for clique generation
• Enables linear-time treewidth computation for planar graphs