An algorithm for an 2 -homological test for the planarity of a graph

Given a finite simple graph Γ, one is able to define the presentation of an associate Coxeter group and construct a CW-complex on which the associated Coxeter group acts. The space is the so-called Davis Complex, denoted and the given graph carries much of the local topological information of the space. This paper summarizes these connections including those between the -homology of and the planarity (or genus) of Γ. The main purpose of this paper is to further investigate this interesting connection between a main topic in geometric group theory (discrete group actions on cellular complexes) and the detection of planar graphs by creating an algorithm we call the -test.