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 $o{m}_t^{+}(G)$ 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 $2\le a\le n$, $n\ge a+3$, it is shown that there exists a connected graph G of order n with $om(G)=a$, $o{m}_t(G)=a+1$ and $o{m}_t^{+}(G)=a+2$.

Keywords:

• Total open monophonic set
• total open monophonic number
• minimal total open monophonic set
• upper total open monophonic number

2010 AMS Subject Classification::

• 05C12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The fourth author's research work was supported by National Board for Higher Mathematics(NBHM), Department of Atomic Energy(DAE), Government of India, Project No. NBHM/R.P.29/2015/Fresh/157.