Minimal total open monophonic sets in graphs

For a connected graph G of order n, a total open monophonic set S of vertices in a graph G is a minimal total open monophonic set if no proper subset of S is a total open monophonic set of G. The upper total open monophonic number of G is the maximum cardinality of a minimal total open monophonic set of G. Certain general properties regarding minimal total open monophonic sets are discussed, and also the upper total open monophonic numbers of certain standard graphs are determined. It is proved that for the Petersen graph G. For integers n and a with , , it is shown that there exists a connected graph G of order n with , and .