On the parameterized complexity of the Edge Monitoring problem

In a graph G=(V,E), a vertex v∈V monitors an edge {u,u′}∈E if {v,u}∈E and {v,u′}∈E. Given an n-vertex graph G=(V,E), in which each edge is contained in at least one triangle, and an integer k, the Edge Monitoring problem consists in finding a set S⊆V of size at most k such that each edge of the graph is monitored by at least one element of S. This problem is known to be NP-hard, even under the unit disk graph. We prove that it is also W[2]-hard when parameterized by k. Using Bidimensionality Theory, we provide an FPT algorithm running in time 2O(k⋅log⁡(maxe∈Eω(e)))⋅n for the weighted version of Edge Monitoring when the input graph is restricted to be apex-minor-free, in particular, it applies to planar graphs, and where we additionally impose each edge e to be monitored at least ω(e) times, and the solution to be contained in a set of selected vertices.