Connected edge monophonic number of a graph

Abstract

For a connected graph G of order n,a set S of vertices is called an edge monophonic set of G if every edge of G lies on a monophonic path joining some pair of vertices in S, and the edge monophonic number m_e(G) is the minimum cardinality of an edge monophonic set. An edge monophonic set S of G is a connected edge mono-phonic set if the subgraph induced by S is connected, and the connected edge monophonic number m_ce(G) is the minimum cardinality of a connected edge monophonic set of G. Graphs of order n with connected edge monophonic number 2, 3 or n are characterized. It is proved that there is no non-complete graph G of order n > 3 with m_e(G) = 3 and m_ce(G) = 3. It is shown that for integers k,l and n with 4 < k < l < n, there exists a connected graph G of order n such that m_e(G) = k and m_ce(G) = l.Also, for integers j,k and l with 4 < j < k < l, there exists a connected graph G such that m_e(G)= j,m_ce(G)= k and g_ce(G) = l,where g_ce(G) is the connected edge geodetic number ofa graph G.