Minimal Total Open Monophonic Sets in Graphs

Abstract

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. Also, it is proved that for a connected graph G of order n, if and only if . For integers n and a with , , it is shown that there exists a connected graph G of order n with , and . It is also proved that for positive integers r, d and , there is a connected graph G with , and