The Conversion Set Problem on Graphs

Abstract

Given a graph G=(V, E) and a threshold function f: V(G) → N, an f-reversible process on G is a dynamical system such that, given an initial vertex labelling c0: V(G)→ {0,1}, every vertex v changes its label if and only if it has at least f(v) neighbors with the opposite label, synchronously in discrete-time steps. An f-conversion set of G is a subset of vertices of G with initial label equal to 1 such that, in an f-reversible process on G, eventually, all vertices reach label 1 and it does not get changed anymore. The conversion set number rf(G) is the minimum cardinality of an f-conversion set of G. The Conversion Set Problem asks whether rf(G) ≤k, which is known to be NP-complete. We prove that it is W[1]-hard when parameterized by the treewidth of G and k together by showing an fpt-reduction from Target Set Selection with the same parameters. We also show a polynomial-time algorithm to determine rf(P) for any path P, which has been left open by Dourado et al. [1]. We also consider a quite similar version on an orientation D=(V, ⃗E) of a graph G =(V, E), that is, the oriented graph obtained from G by choosing one orientation for each edge of G. In this version, a vertex v changes its label if and only if it has at least f(v) incoming neighbors with opposite label. We prove the W[2]-hardness of the Conversion Set Problem for this version parameterized by k, even for an orientation with only one directed cycle and all thresholds equal to 1, and a linear-time algorithm for acyclic orientations.