• THE EDGE –TO–VERTEX MONOPHONIC NUMBER OF A GRAPH

Abstract

For a connected graph G = (V,E) , a monophonic set S  E is called an edge – to – vertex monophonic set if every vertex of G lies on a monophonic path between two vertices in V(S). The edge -to -vertex monophonic number mev(G) of G is the minimum cardinality of its edge – to – vertex monophonic sets. The edge – to – vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. Connected graphs of size q ≥ 3 with edge – to – vertex monophonic number q and q -1 are characterized. It is shown that for positive integers rm, dm and l ≥ 2 with rm < dm  2 rm, there exists a connected graph G with radmG = rm, diammG = dm and mev(G) = l and also shown that for every integers a, b and c with 2  a  b  c, and c  2b – a +1, there exists a connected graph G such that mev(G) = a , gev(G) = b and ߢ (G) = c, where gev(G) is edge – to – vertex geodetic number and ߢ (G) is edge covering number of G.