THE UPPER VERTEX MONOPHONIC NUMBER OF A GRAPH

For any vertex x in a connected graph G of order p ≥ 2, a set S ⊆ V (G) is an x-monophonic set of G if each vertex v ∈ V (G) lies on an x − y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mx(G). An x-monophonic set S is called a minimal x-monophonic set if no proper subset of S is an x-monophonic set. The upper x-monophonic number, denoted by m + (G), is defined as the maximum cardinality of a minimal x-monophonic set of G. We determine bounds for it and find the same for some special classes of graphs. For any two positive integers a and b with 1 ≤ a ≤ b, there exists a connected graph G with mx(G) = a and m + (G) = b for some vertex x in G. Also, it is shown that for any three positive integers a, b and n with a ≥ 2 and a ≤ n ≤ b, there exists a connected graph G with mx(G) = a, m + (G) = b and a minimal x-monophonic set of cardinality n.