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ON THE MONOPHONIC NUMBER OF A GRAPH

For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p 1 are characterized. For every pair a,b of positive integers with 2 � ab, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.