### RESTRAINED DOUBLE MONOPHONIC NUMBER OF A GRAPH

#### Abstract

For a connected graph \(G\) of order at least two, a double monophonic set \(S\) of a graph \(G\) is a restrained double monophonic set if either \(S=V\) or the subgraph induced by \(V-S\) has no isolated vertices. The minimum cardinality of a restrained double monophonic set of \(G\) is the restrained double monophonic number of \(G\) and is denoted by \(dm_{r}(G)\). The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers \(a,\, b,\, c\) with \(3 \leq a \leq b \leq c\), there is a connected graph \(G\) with \(m(G) = a\), \(m_r(G) = b\) and \(dm_{r}(G) = c\), where \(m(G)\) is the monophonic number and \(m_r(G)\) is the restrained monophonic number of a graph \(G\).

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Abdollahzadeh Ahangar H., Samodivkin V., Sheikholeslami S.M. and Abdollah Khodkar. The Restrained Geodetic Number of a Graph. *Bull. Malays. Math. Sci. Soc.*, 2015. Vol. 38. P. 1143–1155. DOI: 10.1007/s40840-014-0068-y

Buckley F., Harary F. *Distance in Graphs*. Addison-Wesley, Redwood City, CA, 1990. 335 p.

Chartrand G., Harary F. and Zhang P. On the geodetic number of a graph. *Networks*, 2002. Vol. 39, No. 1. P. 1–6. DOI: 10.1002/net.10007

Chartrand G., Johns G.L., and Zhang P. On the detour number and geodetic number of a graph. *Ars Combin.*, 2004. Vol. 72. P. 3–15.

Harary F. Graph Theory, Addison-Wesley, 1969.

Harary F., Loukakis E. and Tsouros C. The geodetic number of a graph. *Math. Comput. Modelling*, 1993. Vol. 17, No. 11. P. 89–95. DOI: 10.1016/0895-7177(93)90259-2

Santhakumaran A.P. and Jebaraj T. Double geodetic number of a graph. *Discuss. Math. Graph Theory*, 2012, Vol. 32, No. 1. P. 109–119. DOI: 10.7151/dmgt.1589

Santhakumaran A.P., Titus P. and Ganesamoorthy K. On the monophonic number of a graph. J. *Appl. Math. Inform. *, 2014. Vol. 32, No. 1–2. P. 255–266. DOI: 10.14317/jami.2014.255

Santhakumaran A.P. and Venakata Raghu T. Double monophonic number of a graph. *Int. J. Comput. Appl. Math. *, 2016. Vol. 11, No. 1. P. 21–26. https://www.ripublication.com/ijcam16/ijcamv11n1_03.pdf

Santhakumaran A.P. and Venakata Raghu T. Upper double monophonic number of a graph. *Proyecciones*, 2018, Vol. 37, No. 2. P. 295–304. DOI: 10.4067/S0716-09172018000200295

Santhakumaran A.P. and Venakata Raghu T. Connected double monophonic number of a graph. *Int. J. Math. Comb.*, 2018, Special Issue 1. P. 54–60

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