Subdivision Number of Graphs and Falsity of a conjecture

Abstract

Abstract The domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (where each edge in G can be subdivided atmost once in order to increase the domination number). Arumugam made an interesting conjecture for arbitrary graphs namely for any graph G of order n ≥ 3,3 ≥ sdγ ( G ) ≥ 1. Haynes Hedetniemi, Hedetniemi Jacobs, Knosely and Van der Merwe gave a counter example to the above conjecture by showing that sdγ ( G ) = 4 for the graph K t × K t where t ≥ 4 and modified the conjecture as 4 ≥ sdr ( G ) ≥ 1. In this paper we give an example of a graph with sdγ ( G ) = 5. Restrained subdivision number sdγr(G) is also defined and we show that there exists a graph with sdγr ( G ) = 5.