The P3 infection time is W[1]-hard parameterized by the treewidth

Recent papers investigated the maximum infection times tP3(G), tgd(G) and tmo(G) of the P3 convexity, geodesic convexity and monophonic convexity, respectively. In [4] and [8], it was proved that deciding whether tgd(G)≥2 or tmo(G)≥2 are NP-Complete problems even for bipartite graphs. In [17], it was proved that, in bipartite graphs, deciding whether tP3(G)≥k is polynomial time solvable for k≤4, but is NP-Complete for k≥5. In this paper, we prove that the P3 infection time problem is fixed parameter tractable parameterized by treewidth + k, but is W[1]-hard when parameterized only by the treewidth.