Combining Crown Structures for Vulnerability Measures

Abstract

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. [Katrin Casel et al., 2021] and expand the applicability of crown decomposition techniques.
In summary, we improve the vertex kernel of VI from p³ to 3p², and of wVI from p³ to 3(p² + p^{1.5} p_𝓁), where p_𝓁 < p represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from 𝒪(k²W + kW²) to 3μ(k + √{μ}W), where μ = max(k,W). We also give a combinatorial algorithm that provides a 2kW vertex kernel in fixed-parameter tractable time when parameterized by r, where r ≤ k is the size of a maximum (W+1)-packing. We further show that the algorithm computing the 2kW vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when W = 1, which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a 2k vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.