Combining Crown Structures for Vulnerability Measures

Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number o...

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Bibliographic Details
Published in:Algorithmica Vol. 88; no. 1
Main Authors: Casel, Katrin, Friedrich, Tobias, Niklanovits, Aikaterini, Simonov, Kirill, Zeif, Ziena
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2026
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Combinatorial analysis
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Decomposition
Fields (mathematics)
Graph theory
Mathematics of Computing
Polynomials
Theory of Computation
Crown decomposition
Component order connectivity
Kernelization
Vertex Integrity
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number of vertices that need to be removed to decompose a graph into fragments, but also take into account the size of the largest remaining component. The main focus of our paper is on kernelization algorithms tailored to both measures. We capitalize on the structural attributes inherent in different crown decompositions, strategically combining them to introduce novel kernelization algorithms that advance the current state of the field. In particular, we extend the scope of the balanced crown decomposition provided by Casel et al. [ 1 ] and expand the applicability of crown decomposition techniques. In summary, we improve the vertex kernel of VI from to , and of wVI from to , where represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from to , where . We also give a combinatorial algorithm that provides a 2 kW vertex kernel in fixed-parameter tractable time when parameterized by r , where is the size of a maximum -packing. We further show that the algorithm computing the 2 kW vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when , which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a 2 k vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-025-01348-2