Geodesic
Parameterized Complexity of Safe Set
Journal of Graph Algorithms and Applications, Tome 24 (2020) no. 3, pp. 215-245.

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In this paper we study the problem of finding a small safe set $S$ in a graph $G$, i.e., a non-empty set of vertices such that no connected component of $G[S]$ is adjacent to a larger component in $G - S$. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth $\mathsf{pw}$ and cannot be solved in time $n^{o(\mathsf{pw})}$ unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number $\mathsf{vc}$ unless $\mathrm{PH} = \Sigma^{\mathrm{p}}_{3}$, but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity $\mathsf{nd}$, and (4) it can be solved in time $n^{f(\mathsf{cw})}$ for some double exponential function $f$ where $\mathsf{cw}$ is the clique-width. We also present (5) a faster fixed-parameter algorithm when parameterized by the solution size.
DOI : 10.7155/jgaa.00528
Keywords: safe set, parameterized complexity, vulnerability parameter, pathwidth, clique-width 
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     title = {Parameterized {Complexity} of {Safe} {Set}},
     journal = {Journal of Graph Algorithms and Applications},
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Rémy Belmonte; Tesshu Hanaka; Ioannis Katsikarelis; Michael Lampis; Hirotaka Ono; Yota Otachi. Parameterized Complexity of Safe Set. Journal of Graph Algorithms and Applications, Tome 24 (2020) no. 3, pp. 215-245. doi : 10.7155/jgaa.00528. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00528/

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