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On the General Position Number of Complementary Prisms

Neethu P. K., Chandran S. V. Ullas, Changat Manoj, Klavžar Sandi

Fundamenta Informaticae

2021

Vol. 178, nr 3

267--281

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 G ¯ of G is the graph formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. It is proved that gp(G G ¯ ) ≤ n (G ) + 1 if G is connected and gp(G G ¯ ) ≤ n (G ) if G is disconnected. Graphs G for which gp(G G ¯ ) = n (G ) + 1 holds, provided that both G and G ¯ are connected, are characterized. A sharp lower bound on gp(G G ¯ ) is proved. If G is a connected bipartite graph or a split graph then gp(G G ¯ ) ∈ {n (G ), n (G )+1}. Connected bipartite graphs and block graphs for which gp(G G ¯ ) = 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.