The paper studies the computational complexity of the Vertex Integrity problem from the perspective of parameterized complexity. Vertex integrity is a graph parameter that measures the connectivity of a graph, and it has recently attracted interest in the parameterized complexity community due to its ability to render many hard problems tractable.
The main results of the paper are:
Unweighted Vertex Integrity is shown to be W[1]-hard parameterized by treedepth, resolving an open question from prior work. This is achieved via a reduction from Bounded Degree Vertex Deletion.
Unweighted Vertex Integrity is shown to be W[1]-hard parameterized by the size of the feedback edge set and the maximum degree of the graph. This is proved via a reduction from a variant of the Unary Bin Packing problem.
It is shown that Unweighted Vertex Integrity is FPT parameterized by the max-leaf number of the graph, via a dynamic programming algorithm.
For the weighted version of the problem, it is shown how to extend a previous algorithm for parameter modular width to handle the case of binary-encoded weights, resolving another open problem.
Single-exponential FPT algorithms are provided for Weighted Vertex Integrity parameterized by vertex cover number.
The paper also discusses the relationships between vertex integrity and other well-known structural parameters, providing a more comprehensive understanding of the complexity landscape of this problem.
To Another Language
from source content
arxiv.org
Дополнительные вопросы