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

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 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 \({\rm diam}(G) = 2\), then \({\rm gp}(G)\) is the maximum of \(\omega(G)\) and the maximum order of an induced complete multipartite subgraph of the complement of \(G\). As a consequence, \({\rm gp}(G)\) 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 an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.