The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set S of vertices of a graph G is a general position set if no shortest path in G contains three or more vertices of S. The general position number of G is the number of vertices in a largest general position set. In this paper we investigate the general position numbers of the Mycielskian of graphs. We give tight upper and lower bounds on the general position number of the Mycielskian of a graph G and investigate the structure of the graphs meeting these bounds. We determine this number exactly for common classes of graphs, including cubic graphs and a wide range of trees.

The general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem called the monophonic position problem, obtained by replacing ‘shortest path’ by ‘induced path’. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.

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 Kn□Km if and only if both n and m are odd, and that B wins the game on G∘Kn if and only if either B wins on G or n is even.

The general position number gp( G) of a graph G is the cardinality of a largest set of vertices S such that no element of S lies on a geodesic between two other elements of S. The complementary prism G[Formula: see text] of G is the graph formed from the disjoint union of G and its complement [Formula: see text] by adding the edges of a perfect matching between them. It is proved that gp( G[Formula: see text]) ≤ n( G) + 1 if G is connected and gp( G[Formula: see text]) ≤ n( G) if G is disconnected. Graphs G for which gp( G[Formula: see text]) = n( G) + 1 holds, provided that both G and [Formula: see text] are connected, are characterized. A sharp lower bound on gp( G[Formula: see text]) is proved. If G is a connected bipartite graph or a split graph then gp( G[Formula: see text]) ∈ { n( G), n( G)+1}. Connected bipartite graphs and block graphs for which gp( G[Formula: see text]) = n( G) + 1 holds are characterized. A family of block graphs is constructed in which the gp-number of their complementary prisms is arbitrary smaller than their order.