On the connected monophonic number of a graph. Issue 2 (3rd April 2022) - Ganesamoorthy, K.; Murugan, M.; Santhakumaran, A. P.
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 G [ S ] 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 m c ( G ) . The number of extreme vertices and cut-vertices of G is its extreme-cut order e c ( G ) . A graph G is an extreme-cut connected monophonic graph if m c ( G ) = e c ( G ) . Some interesting results on the extreme-cut connected monophonic graphs G are studied. For positive integers r, d and k ≥ 5 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 2 ≤ d ≤ k − 3 and p − d + k ≥ 0, then there exists an extreme-cut connected monophonic graph G of order p with monophonic diameter d and m c ( G ) = k .