The edge-vertex inequality in a planar graph and a bipartition for the class of all planar graphs

TL;DR

This paper introduces a polynomial-based bipartition of planar graphs into real and complex classes, studies a subclass including grid graphs, and classifies certain small complex planar graphs.

Contribution

It defines a new polynomial criterion for classifying planar graphs and fully characterizes a subclass including grid graphs, advancing understanding of planar graph properties.

Findings

Bipartition of planar graphs into real and complex classes.

Complete recognition of a subclass including grid subgraphs.

Listing of all 2-connected triangle-free complex planar graphs with 7 vertices.

Abstract

For a planar graph with a given f-vector $(f_{0}, f_{1}, f_{2}),$ we introduce a cubic polynomial whose coefficients depend on the f-vector. The planar graph is said to be real if all the roots of the corresponding polynomial are real. Thus we have a bipartition of all planar graphs into two disjoint class of graphs, real and complex ones. As a contribution toward a full recognition of planar graphs in this bipartition, we study and recognize completely a subclass of planar graphs that includes all the connected grid subgraphs. Finally, all the 2-connected triangle-free complex planar graphs of 7 vertices are listed.