Characterization of general position sets and its applications to cographs and bipartite graphs

Abstract

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number $\mathrm{gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If $\mathrm{diam}(G)=2$, then $\mathrm{gp}(G)$ is the maximum of $\omega (G)$ and the maximum order of an induced complete multipartite subgraph of the complement of $G$. As a consequence, $\mathrm{gp}(G)$ of a cograph $G$ can be determined in polynomial time. If $G$ is bipartite, then $\mathrm{gp}(G)\le \alpha (G)$ with equality if $\mathrm{diam}(G)\in \{2,3\}$. A formula for the general position number of the complement of an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.

References in corpus (1)

• On the general position subset selection problem

Cited by in corpus (1)

• The general position problem and strong resolving graph