A planar monomial is by definition an isomorphism class of a finite, planar, reduced rooted tree. If x denotes the tree with a single vertex, any planar monomial is a non-associative product in x relative to m−array grafting. A planar power series f(x) over a field K in x is an infinite sum of K−multiples of planar monomials including the unit monomial represented by the empty tree. For every p...

A graph is fan-crossing free if it has a drawing in the plane so that each edge crossed by independent edges, crossing edges have distinct vertices. On other hand, common vertex, they form fan. Both are prominent examples for beyond-planar graphs. Further well-known classes k-planar, k-gap-planar, quasi-planar, and right angle We use subdivision, node-to-circle expansion path-addition operation...

We prove that triangulated IC-planar and NIC-planar graphs can be recognized in cubic time. A graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. A drawing is IC-planar if, in addition, each vertex is incident to at most one crossing edge and NIC-planar if two pairs of crossing edges share at most one vertex. In a triangulated drawing each face is a triangle. I...

Abstract Methods for improving upper and lower bounds various coverings of planar sets are proposed. New numbers partition constituents presented, suggestions the generalization presented methods offered.