# On the Square Free Detour Number of Windmill Graphs

## DOI:

https://doi.org/10.26713/cma.v14i5.2194## Keywords:

Square free detour number, Connected square free detour number, Vertex square free detour number## Abstract

The set \(S\) of vertices is said to be a square free detour set of \(G^*=( V^*,\,E^* )\) if \(I_{D_{_{ \square f}} } [S]=V^*\). The square free detour number of \(G^*\) is the cardinality of the minimum proper square free detour subset of \(V^*\). The square free detour number \(dn_{\square f} (G^*)\), the connected square free detour number \(cdn_{_{\square f} }(G^*)\) and the vertex square free detour number \(dn_{_{\square f_u} }(G^*)\) of \(G^*\) are defined. Also, we determine the square free detour number, the connected square free detour number and the vertex-square free detour number of windmill graphs.

### Downloads

## References

M.A. Ali and A.A. Ali, The connected detour numbers of special classes of connected graphs, Journal of Mathematics 2019 (2019), Article ID 8272483, 9 pages, DOI: 10.1155/2019/8272483.

S. Arumugam, P. Balakrishnan, A.P. Santhakumaran and P. Titus, The upper connected vertex detour monophonic number of a graph, Indian Journal of Pure and Applied Mathematics 49 (2018), 365 – 379, DOI: 10.1007/s13226-018-0274-7.

G. Chartrand, G.L. Johns and P. Zhang, The detour number of a graph, Utilitas Mathematica 64 (2003), 97 – 113, URL: http://utilitasmathematica.com/index.php/Index/article/view/277.

G. Chartrand, P. Zhang and T.W. Haynes, Distance in graphs – taking the long view, Akce International Journal of Graphs and Combinatorics 1 (2004), 1 – 13, DOI: 10.1080/09728600.2004.12088775.

M. Elakkiya and K. Abhishek, Detour homometric number of a graph, Journal of Physics: Conference Series 1767 (2021), 012009, DOI: 10.1088/1742-6596/1767/1/012009.

J. John and V.R.S. Kumar, The open detour number of a graph, Discrete Mathematics, Algorithms and Applications 13(1) (2021), 2050088, DOI: 10.1142/S1793830920500883.

K.R.S. Narayan and M.S. Sunitha, Detour eccentric and detour distance degree sequences in graphs, Discrete Mathematics, Algorithms and Applications 7(3) (2015), 1550022, DOI: 10.1142/S1793830915500226.

G.P. Pacifica and K.C. Rani, Forcing vertex square free detour number of a graph, Ratio Mathematica 45 (2023), 286 – 295, DOI: 10.23755/rm.v45i0.1030.

J.M. Prabakar and S. Athisayanathan, Connected weak edge detour number of a graph, Mapana Journal of Sciences 15 (2016), 43 – 53, DOI: 10.12723/mjs.38.5.

S.S. Ramalingam, I.K. Asir and S. Athisayanathan, Vertex triangle free detour number of a graph, Mapana Journal of Sciences 15 (2016), 9 – 24, DOI: 10.12723/mjs.38.2.

A.P. Santhakumaran and S. Athisayanathan, On the connected detour number of a graph, Journal of Prime Research in Mathematics 5 (2009), 149 – 170, URL: http://www.sms.edu.pk/jprm/media/pdf/jprm/volume_05/06.pdf

A.P. Santhakumaran, P. Titus and S. Arumugam, The vertex detour number of a graph, AKCE International Journal of Graphs and Combinatorics 4 (2007), 99 – 112, DOI: 10.1080/09728600.2007.12088823.

P. Titus and K. Ganesamoorthy, Upper detour monophonic number of a graph, Electronic Notes in Discrete Mathematics 53 (2016), 331 – 342, DOI: 10.1016/j.endm.2016.05.028.

## Downloads

## Published

## How to Cite

*Communications in Mathematics and Applications*,

*14*(5), 1759–1766. https://doi.org/10.26713/cma.v14i5.2194

## Issue

## Section

## License

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.