The maximum infection time in the geodesic and monophonic convexities

Recent papers investigated the maximum infection time tP3(G) of the P3-convexity (also called maximum time of 2-neighbor bootstrap percolation) and the maximum infection time tmo(G) of the monophonic convexity. In 2014, it was proved that, for bipartite graphs, deciding whether tP3(G)≥k is polynomial time solvable for k≤4, but is NP-complete for k≥5[23]. In 2015, it was proved that deciding whether tmo(G)≥2 is NP-complete even for bipartite graphs [12]. In this paper, we investigate the maximum infection time tgd(G) of the geodesic convexity. We prove that deciding whether tgd(G)≥k is polynomial time solvable for k=1, but is NP-complete for k≥2 even for bipartite graphs. We also present an O(n3m)-time algorithm to determine tgd(G) and tmo(G) in distance-hereditary graphs. For this, we characterize all minimal hull sets of a general graph in the monophonic convexity. Moreover, we improve the complexity of the fastest known algorithm for finding a minimum hull set of a general graph in the monophonic convexity.