Slope Number on Complete Graphs

Objective: A straight line drawing is a mapping of an edge into a straight line segment. The minimum number of distinct slopes used in a straight line drawing of a graph G is called the slope number of the graph G. In this paper the slope number of complete graph is studied elaborately. Methods: This optimization problem is NP-Hard for any arbitrary graph. A canonical way of drawing of a complete graph is an existing one. In present paper, we consider the edges of a complete graph are straight line segments in order to obtain the number of slopes. Findings: This paper interprets the characterization of slopes in complete graph according to an odd and even number of edges and investigated in detail. Moreover, the slope number of a complete graph is compared with the chromatic number of complete graph and the results are observed. Applications/Improvement: Slope number is one of the quality measures of graph drawing. It is used to find out different layout methods for the same graph.