### The Outer Connected Detour Monophonic Number of a Graph

#### Abstract

For a connected graph ???? = (????, ????) of order a set is called a *monophonic set *of ????if every vertex of ????is contained in a monophonic path joining some pair of vertices in ????. The monophonic number (????) of is the minimum cardinality of its monophonic sets. If or the subgraph is connected, then a detour monophonic set of a connected graph is said to be an *outer connected detour monophonic set*of .The *outer** connected**detour**monophonic number* of , indicated by the symbol , is the minimum cardinality of an outer connected detour monophonic set of . The outer connected detour monophonic number of some standard graphs are determined. It is shown that for positive integers , and ???? ≥ 2 with ,there exists a connected graph ????with????????????_{????}???? = _{, }????????????m_{????}???? = and = ????. Also, it is shown that for every pair of integers ????and b with 2 ≤ ???? ≤ ????, there exists a connected graph with and .

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PDF#### References

T. W. Haynes, S. T. Hedetniemi and P. J, Slater, Fundamentals Of Domination In Graphs, Marcel Dekker, New York, (1998).

J. John, The Forcing Monophonic And The Forcing Geodetic Numbers Of A Graph, Indonesian Journal of Combinatorics 4(2), (2020), 114-125.

J. John and S. Panchali 2, The upper monophonic number of a graph, Int. J. Math.Combin. 4, (2010),46 – 52.

P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of A Graph. J. Combin. Math. Combin. Comput., (84), (2013),179-188

DOI: http://dx.doi.org/10.23755/rm.v44i0.921

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