Total Outer Independent Monophonic Number of a Graph

P. Arul Paul Sudhahar1 and A. J. Bertilla Jaushal2

1Department of Mathematics, Rani Anna Govt. College (W), Tirunelveli – 627 008, Tamil Nadu, INDIA. 2Department of Mathematics, Nanjil Catholic College of Arts and Science, Kaliyakkavilai – 629 153, Kanyakumari District, Tamil Nadu, INDIA. Manonmaniam Sundaranar University, Tirunelveli -627 012, Tamil Nadu, INDIA.

DOI : http://dx.doi.org/10.29055/jcms/1138


We initiate the study of total outer independent monophonic in graphs. A set of vertices 𝑀 of a graph 𝐺 is called a total monophonic set if 𝑀 is a monophonic set and its induced subgraph has no isolated vertices. The minimum cardinality of all total monophonic sets of 𝑀 is called the total monophonic number and is denoted by 𝑚𝑡(𝐺). It is shown that For every pair 𝑎,𝑏 of integers with 2≤𝑎≤𝑏 and 𝑏≥2, there exists a connected graph 𝐺 such that 𝑚(𝐺)=𝑎 and 𝑚𝑡𝑜𝑖(𝐺)=𝑏. AMS Subject classification: 05C12.

Keywords :total monophonic set, total monophonic number, total outer independent monophonic set, total outer independent monophonic number.

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