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 gp(G) of G. It is proved that S ⊆ 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 diam(G) = 2, then gp(G) is the maximum of ω(G) and the maximum order of an induced complete multipartite subgraph of the complement of G. As a consequence, gp(G) of a cograph G can be determined in polynomial time. If G is bipartite, then gp(G) ≤ α(G) with equality if diam(G) ∈ {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.1

Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set S of vertices in a graph G is a general position set if no element of S lies on a geodesic between any two other elements of S. The cardinality of a largest general position set is the general position number gpðGÞ of G. The graphs G of order n with gpðGÞ 2 f2, n, n À 1g were already characterized. In this paper, we characterize the classes of all connected graphs of order n ! 4 with the general position number n À 2:

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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.

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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 connected graph G, we say that a set S ⊆ V (G) is an x-position set if for any y ∈ S the shortest x, 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.

The general position problem for graphs was inspired by the no-three-in-line problem and the general position subset selection problem from discrete geometry. A set S of vertices of a graph G is a general position set if no shortest path between two vertices of S contains a third element of S; the general position number of G is the size of a largest general position set. In this note we investigate the general position numbers of the Mycielskian of graphs, which are of interest in chromatic graph theory. 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.

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