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

Abstract

Given a graph G = (V, E), four distinct vertices (Formula presented) and four natural numbers n1, n2, n3, n4such that (Formula presented), we wish to find a partition V1,V2, V3, V4of the vertex set V such that ui (Formula presented) and Vi induces a connected subgraph of G for each i, 1 ≤ i ≤ 4. In this paper we give a simple linear-time algorithm to find such a partition if G is a 4-connected planar graph and u1, u2, us, u4are located on the same face of a plane embedding of G. Our algorithm is based on a “4-canonical decomposition” of G, which is a generahzation of an st-numbering and a “canonical 4-ordering” known in the area of graph drawings.