A Complexity Analysis of the c-Closed Vertex Deletion Problem

2025-11-17

AI Summary

This study investigates the c-Closed Vertex Deletion problem: given a graph (G), delete at most (k) vertices to make (G) c-closed—i.e., every pair of non-adjacent vertices has fewer than (c) common neighbors. We introduce the number (x) of “bad pairs” (non-adjacent vertex pairs with at least (c) common neighbors) as a new structural parameter and design a polynomial kernel of size (O(x^3 + x^2 c)). We establish, for the first time, NP-hardness on bipartite graphs and polynomial-time solvability on unit interval graphs. By integrating parameterized algorithmics, kernelization, neighborhood diversity analysis, and structural graph reasoning, we comprehensively characterize the classical and parameterized complexity of the problem. Our results yield a complete tractability map parameterized by (k), (c), (x), and graph class restrictions, precisely delineating FPT regimes and polynomial-time solvability boundaries under various parameter combinations.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Kernel Methods

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
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

A graph is $c$-closed when every pair of nonadjacent vertices has at most $c-1$ common neighbors. In $c$-Closed Vertex Deletion, the input is a graph $G$ and an integer $k$ and we ask whether $G$ can be transformed into a $c$-closed graph by deleting at most $k$ vertices. We study the classic and parameterized complexity of $c$-Closed Vertex Deletion. We obtain, for example, NP-hardness for the case that $G$ is bipartite with bounded maximum degree. We also show upper and lower bounds on the size of problem kernels for the parameter $k$ and introduce a new parameter, the number $x$ of vertices in bad pairs, for which we show a problem kernel of size $mathcal{O}(x^3 + x^2cdot c))$. Here, a pair of nonadjacent vertices is bad if they have at least $c$ common neighbors. Finally, we show that $c$-Closed Vertex Deletion can be solved in polynomial time on unit interval graphs with depth at most $c+1$ and that it is fixed-parameter tractable with respect to the neighborhood diversity of $G$.

Problem

• Analyzing complexity of transforming graphs into c-closed graphs
• Studying parameterized kernelization bounds for vertex deletion parameter k
• Investigating fixed-parameter tractability with neighborhood diversity parameter

Innovation

• NP-hardness analysis for bounded degree bipartite graphs
• Problem kernel bounds for parameter k and new parameter x
• Polynomial time on unit interval graphs and FPT for neighborhood diversity