Parameterized Complexity of Safe Set
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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 pw and cannot be solved in time n^o(pw) unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number vc unless PH = Σ^p_3, but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity nd, and (4) it can be solved in time n^f(cw) for some double exponential function f where cw is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.
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