Algorithms
The characterization of squaregraphs in terms of distance from a root and links of vertices can be used together with breadth first search as part of a linear time algorithm for testing whether a given graph is a squaregraph, without any need to use the more complex linear-time algorithms for planarity testing of arbitrary graphs.
Several algorithmic problems on squaregraphs may be computed more efficiently than in more general planar or median graphs; for instance, Chepoi, Dragan & Vaxès (2002) and Chepoi, Fanciullini & Vaxès (2004) present linear time algorithms for computing the diameter of squaregraphs, and for finding a vertex minimizing the maximum distance to all other vertices.
Read more about this topic: Squaregraph