The general position number of a graph G is the size of the largest set of vertices S such that no geodesic of G contains more than two elements of S. The monophonic position number of a graph is defined similarly, but with 'induced path' in place of 'geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers, with applications to a realisation result. We then solve a Turán problem for the size of graphs with given order and position numbers and characterise the possible diameters of graphs with given order and monophonic position number. Finally we classify the graphs with given order and diameter and largest possible general position number.

In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex x of a graph G, we say that a set S ⊆ V (G) is an x-position set if for any y ∈ S the shortest 2 Thankachy, Chandran, Tuite, Thomas, Di Stefano and Erskinex, y-paths in G contain no point of S \ {y}. We investigate the largest and smallest orders of maximum x-position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.

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A general position set of a graph G is a set of vertices S in G such that no three vertices from S lie on a common shortest path. In this paper we introduce and study the general position achievement game. The game is played on a graph G by players A and B who alternatively pick vertices of G. A selection of a vertex is legal if has not been selected before and the set of vertices selected so far forms a general position set of G. The player who selects the last vertex wins the game. Playable vertices at each step of the game are described, and sufficient conditions for each of the players to win is given. The game is studied on Cartesian and lexicographic products. Among other results it is proved that A wins the game on K n K m if and only if both n and m are odd, and that B wins the game on G • K n if and only if either B wins on G or n is even.