Characterization of classes of graphs with large general position number

Abstract

Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set $S$ of vertices in a graph $G$ is a general position set if no element of $S$ lies on a geodesic between any two other elements of $S$. The cardinality of a largest general position set is the general position number $\mathrm{gp}(G)$ of $G.$ In \cite{ullas-2016} graphs $G$ of order $n$ with $\mathrm{gp}(G)$ $\in \{2,n,n-1\}$ were characterized. In this paper, we characterize the classes of all connected graphs of order $n\ge 4$ with the general position number $n-2.$