On the General Position Numbers of Maximal Outerplane Graphs

A set R ⊆ V ( G ) of a graph G is a general position set if any triple set R 0 of R is non-geodesic, that is, no vertex of R 0 lies on any geodesic between the other two vertices of R 0 in G . Let R be the set of general position sets of a graph G . The general position number gp ( G ) of a graph G is defined as gp ( G ) = max { | R | : R ∈ R } . In this paper, for an arbitrary maximal outerplane graph G of order at least 7, we prove that gp ( G ) = 3 if and only if G is a straight linear 2-tree. Moreover, we determine the upper bound on the gp-numbers for any maximal outerplane graph and characterize the corresponding extremal graphs.