On the parameterized complexity of the geodesic hull number

Several recent papers obtained complexity results regarding the geodesic hull number hngd(G) of a graph G. In this paper, we prove that determining whether hngd(G)≤k is W[2]-hard parameterized by k in diameter-two graphs and is W[1]-hard parameterized by tw+k, where tw is the treewidth of G. Despite this, for graphs with bounded treewidth tw, we prove that hngd(G) is computable in polynomial time O((tw+2)tw+5n2tw+5).