Non-Separating Planar Graphs

A graph G is a non-separating planar graph if there is a drawing D of G on the plane such that (1) no two edges cross each other in D and (2) for any cycle C in D, any two vertices not in C are on the same side of C in D.Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs.In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain K-1 boolean OR K-4 or K-1 boolean OR K-2,K-3 or K-1,K-1,K-3 as a minor.Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a wheel or it is a graph obtained from the disjoint union of two triangles by adding three vertex-disjoint paths between the two triangles.Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with 3n - 3 edges. Thus, maximal linkless graphs can have significantly fewer edges than maximum linkless graphs; Sachs exhibited linkless graphs with n vertices and 4n - 10 edges (the maximum possible) in 1983.