The Upper Monophonic Hull Number of a
Graph

V. Mary Gleeta

doi:https://doi.org/10.14445/22315373/IJMTT-V60P543

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 60 | Number 5 | Year 2018 | Article Id. IJMTT-V60P543 | DOI : https://doi.org/10.14445/22315373/IJMTT-V60P543

The Upper Monophonic Hull Number of a Graph

V. Mary Gleeta

Abstract

For a connected graph G = (V,E), the monophonic hull number mh(G) of a graph G is the minimum cardinality of a set of vertices whose monophonic convex hull contains all vertices of G. A monophonic hull set S in a connected graph G is called a minimal monophonic hull set of G if no proper subset of S is a monophonic hull set of G. The upper monophonic hull number mh+ (G) of G is the maximum cardinality of a minimal monophonic hull set of G. The upper monophonic hull number of certain classes of graphs are determined. Connected graphs of order p with upper monophonic hull number p or p-1 are characterized. It is shown that for every integer a ≥ 2, there exists a connected graph G with mh(G) =a and mh+ (G) =2a

Keywords

hull number, monophonic hull number, upper monophonic hull number.

References

[1] F.Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, 1990.
[2] G.Chartrand and Ping Zhang, Convex sets in graphs, Congressess Numerantium 136(1999), pp.19-32.
[3] G.Chartrand and P. Zhang, The forcing hull number of a graph,J. Combin Math. Comput. 36(2001), 81-94.
[4] G.Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, (2002) 1-6.
[5] Carmen Hernando, Tao Jiang, Merce Mora, Ignacio. M. Pelayo and Carlos Seara, On the Steiner, geodetic and hull number of graphs, Discrete Mathematics 293 (2005) 139 – 154
[6] M.G.Evertt, S. B. Seidman, The hull number of a graph, Mathematics, 57 (19850 217-223.
[7] Esamel M. paluga, Sergio R. Canoy, Jr, , Monophonic numbers of the join and Composition of connected graphs, Discrete Mathematics 307 (2007) 1146-1154
[8] M.Faber, R.E. Jamison, Convexity in graphs and hypergraphs, SIAM Journal Algebraic Discrete Methods 7(1986) 433-444.
[9] F.Harary, Graph Theory, Addison-Wesley,1969
[10] F.Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput Modeling 17(11)(1993) 89-95.
[11] J.John and S.Panchali, The Upper Monophonic number of a graph, International J.Math.Combin 4(2010),46-52.
[12] J.John and V.Mary Gleeta,Monophonic hull sets in graphs (submitted)
[13] Mitre C. Dourado, Fabio protti and Jayme. L. Szwarcfiter, Algorithmic Aspects of Monophonic Convexity, Electronic Notes in Discrete Mathematics 30(2008) 177-182.
[14] L-D. Tong, The forcing hull and forcing geodetic numbers of graphs Discrete Applied Math.157(2009)1159-1163.

Citation :

V. Mary Gleeta, "The Upper Monophonic Hull Number of a Graph," International Journal of Mathematics Trends and Technology (IJMTT), vol. 60, no. 5, pp. 294-298, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V60P543