A linear algorithm for resource four-partitioning four-connected planar graphs

Given a connected graph G = (V,E), a set V_r ⊆ V of r special vertices, four distinct base vertices u₁, u₂, u₃, u₄ ∈ V and four natural numbers r₁, r₂, r₃, r₄ such that Σ_j=1⁴ r_j = r, we wish to find a partition V₁, V₂, V₃, V₄ of V such that V_i contains u_i and r_i vertices from V_r, and V_i induces a connected subgraph of G for each i, 1 ≤ i ≤ 4. We call a vertex in V_r a resource vertex and the problem above of partitioning vertices of G as the resource four-partitioning problem. In this paper, we give a linear algorithm for finding a resource four-partition of a four-connected planar graph G with base vertices located on the same face of a planar embedding. Our algorithm is based on a 4-canonical decomposition and st-numbering of G.