# The Restrained Edge Monophonic Number of a Graph

## Abstract

A set S of vertices of a connected graph G is a monophonic set if every vertex of G lies on an x−y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). A set S of vertices of a graph G is an edge monophonic set if every edge of G lies on an x − y monophonic path for some elements x and y in S. The minimum cardinality of an edge monophonic set of G is the edge monophonic number of G, denoted by em(G). A set S of vertices of a graph G is a restrained edge monophonic set if either V = S or Sis an edge monophonic set with the subgraph G[V −S] induced by V −S has no isolated vertices. The minimum cardinality of a restrained edge monophonic set of G is the restrained edge monophonic number of G and is denoted by emr(G). It is proved that, for the integers a, b and c with 3 6 a 6 b < c,

there exists a connected graph G having the monophonic number a, the edge monophonic number b and the restrained edge monophonic number c.

## References

[1] F. Buckley and F. Harary. Distance in Graphs. Addison-Wesley, Redwood City, CA, 1990.

[2] F. Buckley, F. Harary, and L. V. Quintas. Extremal Results on the Geodetic Number of a Graph. Scientia A2 (1988), 17-26.

[3] G. Chartrand, F. Harary, and P. Zhang. On the Geodetic Number of a Graph, Networks., 39(1)(2002), 1-6.

[4] G. Chartrand, G.L. Johns, and P. Zhang, On the Detour Number and Geodetic Number of a Graph, Ars Combinatoria, 72(2004), 3-15.

[5] F. Harary, Graph Theory, Addison-Wesley Pub. Co., 1969.

[6] F. Harary, E. Loukakis and C. Tsouros. The Geodetic Number of a Graph, Math. Comput. Modeling, 17(11)(1993), 87-95.

[7] P.A. Ostrand. Graphs with specified radius and diameter, Discrete Math., 4(1)(1973), 71-75.

[8] A. P. Santhakumaran amd J.John. Edge Geodetic Number of a Graph. Journal Discrete Mathematics and Cryptography, 10(3)(2007), 415-432.

[9] A.P. Santhakumaran, P. Titus and K. Ganesamoorthy. On the Monophonic Number of a Graph. J. Appl. Math. & Informatics, 32(1-2)(2014), 255 - 266.

[10] A. P. Santhakumaran, M. Mahendran and P. Titus. The Restrained Edge Geodetic Number of a Graph. International Journal of Computational and Applied Mathematics, 11(1)92016),

9-19.

[11] A. P. Santhakumaran, P. Titus and K. Ganesamoorthy. The Restrained Monophonic Number of a Graph, Communicated.

[2] F. Buckley, F. Harary, and L. V. Quintas. Extremal Results on the Geodetic Number of a Graph. Scientia A2 (1988), 17-26.

[3] G. Chartrand, F. Harary, and P. Zhang. On the Geodetic Number of a Graph, Networks., 39(1)(2002), 1-6.

[4] G. Chartrand, G.L. Johns, and P. Zhang, On the Detour Number and Geodetic Number of a Graph, Ars Combinatoria, 72(2004), 3-15.

[5] F. Harary, Graph Theory, Addison-Wesley Pub. Co., 1969.

[6] F. Harary, E. Loukakis and C. Tsouros. The Geodetic Number of a Graph, Math. Comput. Modeling, 17(11)(1993), 87-95.

[7] P.A. Ostrand. Graphs with specified radius and diameter, Discrete Math., 4(1)(1973), 71-75.

[8] A. P. Santhakumaran amd J.John. Edge Geodetic Number of a Graph. Journal Discrete Mathematics and Cryptography, 10(3)(2007), 415-432.

[9] A.P. Santhakumaran, P. Titus and K. Ganesamoorthy. On the Monophonic Number of a Graph. J. Appl. Math. & Informatics, 32(1-2)(2014), 255 - 266.

[10] A. P. Santhakumaran, M. Mahendran and P. Titus. The Restrained Edge Geodetic Number of a Graph. International Journal of Computational and Applied Mathematics, 11(1)92016),

9-19.

[11] A. P. Santhakumaran, P. Titus and K. Ganesamoorthy. The Restrained Monophonic Number of a Graph, Communicated.

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## Published

2016-12-30

## Issue

## Section

Чланци