A characterization of general position sets in graphs
Bijo S. Anand, Ullas Chandran S.V., Manoj Changat, Sandi Klavžar, Elias John Thomas · arXiv (Cornell University) · 2018
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 ${\rm gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is a general position set 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 ${\rm diam}(G) = 2$, then ${\rm gp}(G)$ is the maximum of the clique number of $G$ and the maximum order of an induced complete multipartite subgraph of the complement of $G$. As a consequence, ${\rm gp}(G)$ of a cograph $G$ can be determined in polynomial time. If $G$ is bipartite, then ${\rm gp}(G) \leq \alpha(G)$ with equality if ${\rm diam}(G) \in \{2,3\}$. A formula for the general position number of the complement of a bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.