Cluster Vertex Deletion on Chordal Graphs

2026-04-22

AI Summary

This work addresses the Cluster Vertex Deletion problem on chordal graphs—namely, removing the minimum number of vertices to transform the graph into a disjoint union of cliques. Despite its NP-hardness in general, we present the first polynomial-time exact algorithm for chordal graphs by leveraging their clique tree structure. Our approach formulates a dynamic programming framework over the clique tree and reduces the computation of optimal subproblem values to minimizing a submodular set function. This method not only yields an efficient solution for chordal graphs but also resolves several open questions recently posed in the literature, offering both theoretical insights and algorithmic advancements.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Combinatorial OptimizationData Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a polynomial-time algorithm for the cluster vertex deletion problem on chordal graphs, resolving an open question posed in different contexts by Cao et al. [Theoretical Computer Science, 2018], Aprile et al. [Mathematical Programming, 2023], Chakraborty et al. [Discrete Applied Mathematics, 2024], and Hsieh et al. [Algorithmica, 2024]. We use dynamic programming over clique trees and reduce the computation of the optimal subproblem value to the minimization of a submodular set function.

Problem

• Cluster Vertex Deletion
• Chordal Graphs
• Graph Modification
• Clique
• Vertex Deletion

Innovation

• cluster vertex deletion
• chordal graphs
• dynamic programming
• clique trees
• submodular minimization