Abstract
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 of G. The graphs G of order n with were already characterized. In this paper, we characterize the classes of all connected graphs of order with the general position number
Highlights
The general position problem in graphs was introduced by P
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
In [7] it is proved that for a connected graph of order n, the complete graph of order n is the only graph with the largest general position number n; and gp(G) = n − 1 if and only if G = K1 + j mjKj with mj ≥ 2 or G = Kn − {e1, e2, . . . , ek} with 1 ≤ k ≤ n − 2, where ei’s all are edges in Kn which are incident to a common vertex v
Summary
The general position problem in graphs was introduced by P. In [7] it is proved that for a connected graph of order n, the complete graph of order n is the only graph with the largest general position number n; and gp(G) = n − 1 if and only if G = K1 + j mjKj with mj ≥ 2 or G = Kn − {e1, e2, . In the same paper it is proved that the general position problem is NP-complete for arbitarary graphs. The gp-number of graphs of diameter 2, cographs, graphs with at least one universal vertex, bipartite graphs and their complements were obtained. In [8] a sharp lower bound on the gp-number is proved for Cartesian products of graphs. Recent developments on general position number can be seen in [9]
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