Properties of π-skew Graphs with Applications

Abstract

The skewness of a graph G, denoted by sk(G), is the minimum number of edges in G whose removal results in a planar graph. It is an important parameter that measures how close a graph is to planarity, and it is complementary, and computationally equivalent, to the Maximum Planar Subgraph Problem. For any connected graph G on p vertices and q edges with girth g, one can easily verify that sk(G) ≥ π(G), where $$\pi(G)=\lceil q-\frac{g}{g-2}(p-2)\rceil$$ , and the graph G is said to be π-skew if equality holds. The concept of π-skew was first proposed by G. L. Chia and C. L. Lee. The π-skew graphs with girth 3 are precisely the graphs that contain a triangulation as a spanning subgraph. The purpose of this paper is to explore the properties of π-skew graphs. Some families of π-skew graphs are obtained by applying these properties, including join of two graphs, complete multipartite graphs and Cartesian product of two graphs. We also discuss the threshold for the existence of a spanning triangulation. Among other results some sufficient conditions regarding the regularity and size of a graph, which ensure a spanning triangulation, are given.