The Forcing Monophonic Hull Number of a Graph

J. John; V. Mary Gleeta

doi:https://doi.org/10.14445/22315373/IJMTT-V3I2P501

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 3 | Issue 2 | Year 2012 | Article Id. IJMTT-V3I2P501 | DOI : https://doi.org/10.14445/22315373/IJMTT-V3I2P501

The Forcing Monophonic Hull Number of a Graph

J. John, V. Mary Gleeta

Citation :

J. John, V. Mary Gleeta, "The Forcing Monophonic Hull Number of a Graph," International Journal of Mathematics Trends and Technology (IJMTT), vol. 3, no. 2, pp. 43-46, 2012. Crossref, https://doi.org/10.14445/22315373/IJMTT-V3I2P501

Abstract

For a connected graph G = (V, E), let a set M be a minimum monophonic hull set of G. A subset T  M is called a forcing subset for M if M is the unique minimum monophonic hull set containing T. A forcing subset for M of minimum cardinality is a minimum forcing subset of M. The forcing monophonic hull number of M, denoted by fmh(M), is the cardinality of a minimum forcing subset of M. The forcing monophonic hull number of G, denoted by fmh(G), is fmh(G)=min{fmh(M)}, where the minimum is taken over all minimum monophonic hull sets in G. Some general properties satisfied by this concept are studied. The forcing monophonic hull numbers of certain classes of graphs are determined. It is shown that, for every pair a, b of integers with 0 ≤ a ≤ b and b ≥ 2, there exists a connected graph G such that fmh(G) = a and mh(G) = b.

Keywords

hull number, monophonic hull number, forcing hull number, forcing 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.