Extreme outer connected monophonic graphs

Document Type : Original paper

Authors

1 Department of Mathematics, Coimbatore Institute of Technology (Government Aided Autonomous Institution) Coimbatore - 641 014, India

2 Department of Mathematics, Coimbatore Institute of Technology, Coimbatore

Abstract

For a connected graph $G$ of order at least two, a  set $S$ of vertices in a graph $G$ is said to be an \textit{outer connected monophonic set} if $S$ is a monophonic set of $G$ and either $S=V$ or the subgraph induced by $V-S$ is connected. The minimum cardinality of an outer connected monophonic set of $G$ is the \textit{outer connected monophonic number} of $G$ and is denoted by $m_{oc}(G)$.  The number of extreme vertices in $G$ is its \textit{extreme order} $ex(G)$.  A graph $G$ is said to be an \textit{extreme outer connected monophonic graph} if $m_{oc}(G)$ = $ex(G)$.  Extreme outer connected monophonic graphs of order $p$ with outer connected monophonic number $p$ and extreme outer connected monophonic graphs of order $p$ with outer connected monophonic number $p-1$ are characterized.  It is shown that for every pair $a, b$ of integers with $0 \leq a \leq b$ and $b \geq 2$, there exists a connected graph $G$ with $ex(G) = a$ and $m_{oc}(G) = b$. Also, it is shown that  for positive integers $r,d$ and $k \geq 2$ with $r < d$, there exists an extreme outer connected monophonic graph $G$ with monophonic radius $r$, monophonic diameter $d$ and outer connected monophonic number $k$.

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References

1] H. Abdollahzadeh Ahangar, F. Fujie-Okamoto, and V. Samodivkin, On the forcing connected geodetic number and the connected geodetic number of a graph, Ars Combin. 126 (2016), 323–335.
[2] H. Abdollahzadeh Ahangar, S. Kosari, S.M. Sheikholeslami, and L. Volkmann, Graphs with large geodetic number, Filomat 29 (2015), no. 6, 1361–1368.
[3] H. Abdollahzadeh Ahangar and M. Najimi, Total restrained geodetic number of graphs, Iran. J. Sci. Technol. Trans. A Sci. 41 (2017), no. 2, 473–480.
[4] H. Abdollahzadeh Ahangar, V. Samodivkin, S.M. Sheikholeslami, and A. Khodkar, The restrained geodetic number of a graph, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 3, 1143–1155.
[5] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, 1990.
[6] F. Buckley, F. Harary, and L.V. Quintas, Extremal results on the geodetic number of a graph, Scientia A 2 (1998), 17–26.
[7] G. Chartrand, F. Harary, and P. Zhang, On the geodetic number of a graph, Networks 39 (2002), no. 1, 1–6.
[8] G. Chartrand, E.M. Palmer, and P. Zhang, The geodetic number of a graph, a survey, Cong. Numer. 156 (2002), 37–58.
[9] E.R. Costa, M.C. Dourado, and R. M. Sampaio, Inapproximability results related to monophonic convexity, Discrete Appl. Math. 197 (2015), 70–74.
[10] M.C. Dourado, F. Protti, and J.L. Szwarcfiter, Algorithmic aspects of monophonic convexity, Electronic Notes in Discrete Math. 30 (2008), 177–182.
[11] , Complexity results related to monophonic convexity, Discrete Appl. Math. 158 (2010), no. 12, 1268–1274.
[12] K. Ganesamoorthy and D. Jayanthi, Extreme outer connected geodesic graphs, (Communicated).
[13] , The outer connected geodetic number of a graph, Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci. 91 (2021), 195–200.
[14] K. Ganesamoorthy and S. Lakshmi Priya, The outer connected monophonic number of a graph, Ars Combin. 153 (2020), 149–160.
[15] F. Harary, Graph Theory, Addison-Wesley, 1969.
[16] F. Harary, E. Loukakis, and C. Tsouros, The geodetic number of a graph, Mathl. Comput. Modeling 17 (1993), no. 11, 89–95.
[17] J. John and D. Stalin, Distinct edge geodetic decomposition in graphs, Commun. Comb. Optim. 6 (2021), no. 2, 185–196.
[18] E.M. Paluga and S.R. Canoy Jr, Monophonic numbers of the join and composition of connected graphs, Discrete Math. 307 (2007), no. 9-10, 1146–1154.
[19] V. Samodivkin, On the edge geodetic and edge geodetic domination numbers of a graph, Commun. Comb. Optim. 5 (2020), no. 1, 41–54.
[20] A.P. Santhakumaran and P. Titus, Monophonic distance in graphs, Discrete Math. Algorithms Appl. 3 (2011), no. 2, 159–169.
[21] , A note on “ monophonic distance in graphs”, Discrete Math. Algorithms Appl. 4 (2012), no. 2, Article ID: 1250018.
[22] A.P. Santhakumaran, P. Titus, and K. Ganesamoorthy, On the monophonic number of a graph, J. Appl. Math. & Informatics 32 (2014), no. 1 2, 255–266.
 
Communications in Combinatorics and Optimization
Volume 7, Issue 2
December 2022
Pages 211-226

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