Locating-dominating number of certain infinite families of convex polytopes with applications

Abstract

A convex hull of finitely many points in the Euclidean space $R^d$ is known as a convex polytope. Graphically, they are planar graphs i.e. embeddable on $R^2$. Minimum dominating sets possess diverse applications in computer science and engineering. Locating-dominating sets are a natural extension of dominating sets. Studying minimizing locating-dominating sets of convex polytopes reveal interesting distance-dominating related topological properties of these geometrical planar graphs. In this paper, exact value of the locating-dominating number is shown for one infinite family of convex polytopes. Moreover, tight upper bounds on ${\gamma }_{l-d}$ are shown for two more infinite families. Tightness in the upper bounds is shown by employing an updated integer linear programming (ILP) model for the locating-dominating number ${\gamma }_{l-d}$ of a fixed graph. Results are explained with help of some examples. The second part of the paper solves an open problem in Khan (2023) [28] which asks to find a domination-related parameter which delivers a correlation coefficient of $\rho >0.9967$ with the total π-electronic energy of lower benzenoid hydrocarbons. We show that the locating-dominating number ${\gamma }_{l-d}$ delivers such a strong prediction potential. The paper is concluded with putting forward some open problems in this area.

Keywords: 05C10; 05C69; 90C05; 92E10; Convex polytope; Domination number; Graph; ILP model; Locating-dominating number; Structure-property modeling.