Problem
Research questions and friction points this paper is trying to address.
clique transversal
W[1]-hardness
parameterized complexity
maximal clique
W[1]-Hardness of Upper Clique Transversal
🤖 AI Summary
This study investigates the parameterized complexity of determining whether a graph contains an inclusion-wise minimal clique transversal of size at least $k$. By applying parameterized complexity theory and reduction techniques, we rigorously prove that this decision problem is W[1]-hard when parameterized by $k$. This result establishes the computational hardness boundary of the problem within the parameterized framework, indicating that a fixed-parameter tractable (FPT) algorithm is unlikely to exist. Consequently, our findings provide a crucial theoretical foundation for research on the computational complexity of related graph-theoretic problems.
📝 Abstract
A clique transversal of a graph is a set of vertices intersecting every maximal clique. We prove that deciding whether a graph has an inclusion-wise minimal clique transversal of size at least $k$ is W[1]-hard when parameterized by $k$.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Clique Transversal
W[1]-Hardness
Parameterized Complexity
Maximal Clique
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