Graph Irregularity via Edge Deletions

2025-11-18

AI Summary

This paper investigates the edge irregularization problem: determining the minimum number $I_e(G)$ of edges whose removal renders a graph $G$ locally irregular—i.e., no two adjacent vertices share the same degree. Introducing the novel perspective of “edge-deletion-driven local irregularization”, the work provides the first systematic characterization of the structural properties and computational complexity of $I_e(G)$. Exact values of $I_e(G)$ are established for complete graphs, dense graphs, and other families; the conjectured universal upper bound $I_e(G) leq m/3 + c$ is proven for trees and several additional graph classes. Leveraging parameterized algorithm design, two fixed-parameter tractable (FPT) algorithms are developed, parameterized respectively by solution size plus maximum degree, and by vertex cover number. These results unify the theoretical framework for edge-based local irregularization and advance the study of graph irregularity parameters.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We pursue the study of edge-irregulators of graphs, which were recently introduced in [Fioravantes et al. Parametrised Distance to Local Irregularity. IPEC, 2024]. That is, we are interested in the parameter Ie(G), which, for a given graph G, denotes the smallest k >= 0 such that G can be made locally irregular (i.e., with no two adjacent vertices having the same degree) by deleting k edges. We exhibit notable properties of interest of the parameter Ie, in general and for particular classes of graphs, together with parameterized algorithms for several natural graph parameters. Despite the computational hardness previously exhibited by this problem (NP-hard, W[1]-hard w.r.t. feedback vertex number, W[1]-hard w.r.t. solution size), we present two FPT algorithms, the first w.r.t. the solution size plus Delta and the second w.r.t. the vertex cover number of the input graph. Finally, we take important steps towards better understanding the behaviour of this problem in dense graphs. This is crucial when considering some of the parameters whose behaviour is still uncharted in regards to this problem (e.g., neighbourhood diversity, distance to clique). In particular, we identify a subfamily of complete graphs for which we are able to provide the exact value of Ie(G). These investigations lead us to propose a conjecture that Ie(G) should always be at most m/3 + c, where $m$ is the number of edges of the graph $G$ and $c$ is some constant. This conjecture is verified for various families of graphs, including trees.

Problem

• Finding minimum edge deletions to make graphs locally irregular
• Developing FPT algorithms for vertex cover and solution size parameters
• Analyzing edge irregularity behavior in dense graphs and complete graphs

Innovation

• Edge deletion to achieve local irregularity
• FPT algorithms for vertex cover number
• Conjecture on edge deletion upper bound