On the Connected Monophonic Number of a Graph

Abstract

For a connected graph G of order at least two, a connected monophonic set of G is a monophonic set S such that the subgraph induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by . The number of extreme vertices and cut-vertices of G is its extreme-cut order . A graph G is an extreme-cut connected monophonic graph if . Some interesting results on the extreme-cut connected monophonic graphs G are studied. For positive integers r, d and with r<d, there exists an extreme-cut connected monophonic graph G with monophonic radius r, monophonic diameter d and the connected monophonic number k. Also if p, d and k are positive integers such that and , then there exists an extreme-cut connected monophonic graph G of order p with monophonic diameter d and .