The Connected and Forcing Connected Restrained Monophonic Numbers of a Graph
DOI:
https://doi.org/10.22199/issn.071762795390Keywords:
monophonic set, restrained monophonic set, connected restrained monophonic set, forcing connected restrained monophonic setAbstract
For a connected graph G = (V,E) of order at least two, a connected restrained monophonic set S of G is a restrained monophonic set such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected restrained monophonic set of G is the connected restrained monophonic number of G and is denoted by mcr(G). We determine bounds for it and find the same for some special classes of graphs. It is shown that, if a, b and p are positive integers such that 3 ≤ a ≤ b ≤ p, p−1 6= a, p−1 6= b, then there exists a connected graph G of order p, mr(G) = a and mcr(G) = b. Also, another parameter forcing connected restrained monophonic number fcrm(G) of a graph G is introduced and several interesting results and realization theorems are proved.
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References
H. Abdollahzadeh Ahangar, V. Samodivkin, S.M. Sheikholeslami and Abdollah Khodkar, The Restrained Geodetic Number of a Graph, Bull. Malays. Math. Sci. Soc., 2015, 38, 1143 1155.
M.C. Dourado, F. Protti and J.L. Szwarcfiter, Algorithmic Aspects of Monophonic Convexity, Electronic Notes in Discrete Mathematics, 2008, 30, 177182.
F. Buckley and F. Harary, Distance in Graphs, AddisonWesley, Redwood City, CA, (1990).
F. Buckley, F. Harary, and L. V. Quintas, Extremal Results on the Geodetic Number of a Graph, Scientia 1988, A2, 1726.
G. Chartrand, F. Harary, and P. Zhang, On the Geodetic Number of a Graph, Networks, 2002, 39, 16.
G. Chartrand, G.L. Johns, and P. Zhang, On the Detour Number and Geodetic Number of a Graph, Ars Combinatoria, 2004, 72, 315.
F. Harary, Graph Theory, AddisonWesley, 1969.
F. Harary, E. Loukakis and C. Tsouros, The Geodetic Number of a Graph, Math. Comput. Modeling 1993, 17(11), 8795.
E.M. Palugaa and S.R. Canoy, Monophonic numbers of the join and composition of connected graphs, Discrete Mathematics, 2007, 307, 11461154.
A.P. Santhakumaran, M. Mahendran and K. Ganesamoorthy, The Restrained Open Monophonic Number of a Graph, Ars Combinatoria, 2021,155, 193205
A.P. Santhakumaran, P. Titus, K. Ganesamoorthy, On the Monophonic Number of a Graph, J. Appl. Math. & Informatics, 2014, 32(12), 255 266.
A. P. Santhakumaran, P. Titus and K. Ganesamoorthy, The Restrained Monophonic Number of a Graph, TWMS J. App. and Eng. Math., Accepted.
A.P. Santhakumaran, P. Titus and K. Ganesamoorthy, Extreme Restrained Monophonic Graphs, Ars Combinatoria, 2019, 145, 207223.
P. Titus and K. Ganesamoorthy, The Connected Monophonic Number of a Graph, Graphs and Combinatorics, 2014, 30, 237 245.
A. P. Santhakumaran, T. Venkata Raghu and K. Ganesamoorthy, Minimal Restrained Monophonic Sets in Graphs, TWMS J. App. and Eng. Math., 2021, 11(3), 762771.
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