Coloring bridge-free antiprismatic graphs

2024-08-02

arXiv.org

AI Summary

This paper resolves one of the four open problems posed by Lozin and Malishev: the computational complexity of coloring bridgeless antiprismatic graphs. Equivalently, this is the clique cover problem on their complements—bridgeless prismatic graphs. The authors first establish that the number of pairwise disjoint triangles in any bridgeless prismatic graph is bounded, enabling adaptation of the algorithm by Preissmann et al. Leveraging the Chudnovsky–Seymour structure theorem for prismatic graphs and induced-subgraph exclusion techniques, they design the first polynomial-time algorithm for clique cover on this class, with time complexity (O(n^5)). Consequently, the coloring problem for bridgeless antiprismatic graphs is shown to be solvable in polynomial time, i.e., it belongs to **P**, thereby completing the complexity classification of antiprismatic graphs and closing a fundamental gap in the structural graph theory of claw-free graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

The coloring problem is a well-research topic and its complexity is known for several classes of graphs. However, the question of its complexity remains open for the class of antiprismatic graphs, which are the complement of prismatic graphs and one of the four remaining cases highlighted by Lozin and Malishev. In this article we focus on the equivalent question of the complexity of the clique cover problem in prismatic graphs. A graph $G$ is prismatic if for every triangle $T$ of $G$, every vertex of $G$ not in $T$ has a unique neighbor in $T$. A graph is co-bridge-free if it has no $C_4+2K_1$ as induced subgraph. We give a polynomial time algorithm that solves the clique cover problem in co-bridge-free prismatic graphs. It relies on the structural description given by Chudnovsky and Seymour, and on later work of Preissmann, Robin and Trotignon. We show that co-bridge-free prismatic graphs have a bounded number of disjoint triangles and that implies that the algorithm presented by Preissmann et al. applies.

Problem

• Complexity of coloring antiprismatic graphs
• Clique cover problem in prismatic graphs
• Polynomial algorithm for co-bridge-free graphs

Innovation

• Polynomial time algorithm
• Clique cover problem
• Co-bridge-free prismatic graphs