The maximum infection time of the P3 convexity in graphs with bounded maximum degree

Abstract Recent papers investigated the maximum infection times t P 3 ( G ) , t g d ( G ) and t m o ( G ) of the P 3 convexity, geodesic convexity and monophonic convexity, respectively. In Benevides et al. (2016) and Costa et al. (2015), it was proved that deciding whether t g d ( G ) ≥ 2 or t m o ( G ) ≥ 2 are NP-Complete problems even for bipartite graphs. In Marcilon et al. (2014), it was proved that, in bipartite graphs, deciding whether t P 3 ( G ) ≥ k is polynomial time solvable for k ≤ 4 , but is NP-Complete for k ≥ 5 . In this paper, we prove that, in grid graphs with maximum degree 3, t P 3 ( G ) is NP-hard (answering a question of Benevides et al. (2015)), but is polynomial time solvable if the grid graph is solid. Moreover, we prove that deciding whether t P 3 ( G ) ≥ n − k is polynomial time solvable for any fixed k , but is NP-Complete for k = n e − 4 for every fixed 0 e ≤ 1 , generalizing the main result of Benevides et al. (2015). Finally, for any fixed Δ , we prove that, in graphs with bounded maximum degree Δ , deciding whether t P 3 ( G ) ≥ k = Θ ( log n ) is polynomial time solvable if Δ ≤ 3 , but is NP-Complete if Δ ≥ 4 .