Some structural, metric and convex properties of the boundary of a graph

Let u, v be two vertices of a connected graph G. The vertex v is said to be a boundary vertex of u if no neighbor of v is further away from u than v. The boundary of a graph is the set of all its boundary vertices. In this work, we present a number of properties of the boundary of a graph under different points of view: (1) a realization theorem involving different types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary; (2) the contour is a monophonic set; and (3) the cardinality of the boundary is an upper bound for both the metric dimension and the determining number of a graph.