Parameterized Algorithms for Power Edge Set and Zero Forcing Set

Abstract

In this article, we study the parameterized complexity of the Power Edge Set problem (abbreviated as PES). In PES, we are given a graph G and an integer k, and the goal is to find a set S⊆E(G) of size at most k such that Smonitors all the vertices of G. An edge set S is said to monitor the vertex set V(G) if, starting with the endpoints of the edges in S (which are initially considered monitored), the entire vertex set can be monitored by repeatedly applying the following rule: if there exists a monitored vertex with exactly one unmonitored neighbor, then that neighbor becomes monitored. The parameterized complexity of this problem was initiated by Darties et al. (Journal of Discrete Algorithms, 2018), who explicitly posed the question of whether PES is fixed-parameter tractable (FPT)—that is, whether it admits an algorithm with running time f(k)·nO(1). Cazals et al. (IWOCA, 2019) subsequently showed that a precolored variant of the problem is W[2]-hard. While their introduction mentions that PES is W[2]-hard, the reduction only applies to the precolored version. In this paper, we clarify the complexity of PES and show that both PES and its natural variant, Zero Forcing Set, are indeed fixed-parameter tractable. Our results include an FPT algorithm for PES parameterized by the treewidth of the input graph, thereby improving upon a previously known XP algorithm for this parameter. Furthermore, we present efficient FPT algorithms for PES when parameterized by vertex cover number and by neighborhood diversity. © 2025 Elsevier B.V., All rights reserved.