The Total Open Monophonic Number of a Graph

Abstract

For a connected graph G of order n >_- 2, a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G).  A  monophonic set of cardinality m(G) is called a m-set of G. A set S of vertices  of a connected graph G is an open monophonic set of G if for each vertex v  in G, either v is an extreme vertex of G and v ˆˆ? S, or v is an internal vertex of a x-y monophonic path for some x, y ˆˆ? S. An open monophonic set of minimum cardinality is a minimum open monophonic set and this cardinality is the open monophonic number, om(G). A connected open monophonic set of G is an open monophonic set S such that the subgraph < S > induced by S is connected. The minimum cardinality of a connected open monophonic set of G is the connected open monophonic number of G and is denoted by omc(G). A total open monophonic set of a graph G is an open monophonic set S such that the subgraph < S > induced by S contains no isolated vertices. The minimum cardinality of a total open monophonic set of G is the total open monophonic number of G and is denoted by omt(G). A total open monophonic set of cardinality omt(G) is called a omt-set of G. The total open monophonic  numbers of certain standard graphs are determined. Graphs with total open monphonic number 2 are characterized. It is proved that if G is a connected graph such that omt(G) = 3 (or omc(G) = 3), then G = K3 or G contains exactly two extreme vertices. It is proved that for any integer n  3, there exists a connected graph G of order n such that om(G) = 2, omt(G) = omc(G) = 3. It is proved that for positive integers r, d and k  4 with 2r, there exists a connected graph of radius r, diameter d and total open monophonic number k. It is proved that for positive integers a, b, n with 4 <_ a<_ b <_n, there exists  a connected graph G of order n such that omt(G) = a and omc(G) = b.