Parameterized Complexity of Safe Set

Abstract

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. 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 and cannot be solved in time unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number unless, but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity, and (4) it can be solved in time for some double exponential function f where is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.