The Price of Connectivity Augmentation on Planar Graphs

Given two classes of graphs, 𝒢₁ ⊆ 𝒢₂, and a c-connected graph G ∈ 𝒢₁, we wish to augment G with a smallest cardinality set of new edges F to obtain a k-connected graph G' = (V,E∪ F) ∈ 𝒢₂. In general, this is the c → k connectivity augmentation problem. Previous research considered variants where 𝒢₁ = 𝒢₂ is the class of planar graphs, plane graphs, or planar straight-line graphs. In all three settings, we prove that the c → k augmentation problem is NP-complete when 2 ≤ c < k ≤ 5. However, the connectivity of the augmented graph G' is at most 5 if 𝒢₂ is limited to planar graphs. We initiate the study of the c → k connectivity augmentation problem for arbitrary k ∈ ℕ, where 𝒢₁ is the class of planar graphs, plane graphs, or planar straight-line graphs, and 𝒢₂ is a beyond-planar class of graphs: 𝓁-planar, 𝓁-plane topological, or 𝓁-plane geometric graphs. We obtain tight bounds on the tradeoffs between the desired connectivity k and the local crossing number 𝓁 of the augmented graph G'. We also show that our hardness results apply to this setting. The connectivity augmentation problem for triangulations is intimately related to edge flips; and the minimum augmentation problem to the flip distance between triangulations. We prove that it is NP-complete to find the minimum flip distance between a given triangulation and a 4-connected triangulation, settling an open problem posed in 2014, and present an EPTAS for this problem.