The Connected Monophonic Number of a Graph

For a connected graph G = (V, E), a monophonic set of G is a set M � V (G) such that every vertex of G is contained in a monophonic path joining some pair of vertices in M. The monophonic number m (G) of G is the minimum order of its monophonic sets and any monophonic set of order m (G) is a minimum monophonic set of G. A connected monophonic set of a graph G is a monophonic set M such that the subgraph < M > induced by M 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). Connected graphs of order p with connected monophonic number 2 and p are characterized. It is shown that for any positive integers 2 � a < b � c, there exists a connected graph G with m (G) = a, m c (G) = b and g c (G), where g c (G) is the connected geodetic number of a graph. AMS SUBJECT CLASSICATION: 05C12.