Minimal total open monophonic sets in graphs. Issue 1 (2nd January 2022) - Santhakumaran, A. P.; Mahendran, M.; Simon Raj, F.; Ganesamoorthy, K.
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 o m t + ( G ) 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 o m t + ( G ) = 4 for the Petersen graph G . For integers n and a with 2 ≤ a ≤ n, n ≥ a + 3, it is shown that there exists a connected graph G of order n with o m ( G ) = a, o m t ( G ) = a + 1 and o m t + ( G ) = a + 2 .