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 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.

A combinatorial branch and bound for the safe set problem

“…On planar graphs and split graphs, the problem is ‐hard even in the unweighted case: an approximation‐preserving reduction from the Minimum Vertex Cover Problem [4] implies that approximation guarantees better than 1.3606 are possible only if [1]. Belmonte et al [3] analyze the parameterized complexity of the unweighted problem. The variant of the problem known as Connected Safe Set Problem ( CSSP ) imposes on the solution the additional constraint to induce a connected subgraph.…”

“…FIGURE3 Example of graph manipulations: removing each of the three edges (2, 3), (6, 7) or (12, 13) (in red) or replacing edge (9, 10) (also in red) with (1, 10) (in dotted blue) yields a relaxation of the original problem…”