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

N.E Johnwin Beaula, S Joseph Robin

#### 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 setof .The outer connecteddetourmonophonic 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 .

#### Keywords

chord, monophonic path, monophonic number, detour monophonic path, detour monophonic number, outer connected detour monophonic number

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#### References

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

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Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.