Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs

Canan Çiftçi

doi:10.33401/fujma.1862064

Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs

Abstract

A set $S \subseteq V(G)$ is a disjunctive total dominating set of $G$ if each vertex $u \in V(G)$ is either adjacent to a vertex in $S$ or is at distance two from at least two vertices in $S$. The minimum cardinality of a disjunctive total dominating set is the disjunctive total domination number. The disjunctive total domination subdivision number is the minimum number of edges that must be subdivided (each edge can be subdivided at most once) to increase the disjunctive total domination number of $G$. In this paper, we investigate the disjunctive total domination and its corresponding subdivision number for various cycle related graphs, including generalized fan, claw-free, and $k$-pyramid graphs. Furthermore, we extend our study to wheel related structures such as gear, helm, and web graphs.

Keywords

References

[1] W. Goddard, M.A. Henning, and C.A. McPillan, The disjunctive domination number of a graph, Quaest. Math., 37(4) (2014), 547–561. $ \href{https://doi.org/10.2989/16073606.2014.894688}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84926223162?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000346905700006}{\mbox{[Web of Science]}} $
[2] M. A. Henning and V. Naicker, Disjunctive total domination in graphs, J. Comb. Optim., 31 (3) (2016), 1090–1110. $ \href{https://doi.org/10.1007/s10878-014-9811-4}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84910599692?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000372282300010} {\mbox{[Web of Science]}} $
[3] V. Naicker and M. A. Henning, Graphs with large disjunctive total domination number, Discrete Math. Theor. Comput. Sci., 17 (1) (2015), 255–281. $ \href{https://doi.org/10.46298/dmtcs.2104 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84928409100?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000365862100001}{\mbox{[Web of Science]}} $
[4] C. Çiftçi, Disjunctive total domination of some shadow distance graphs, Fundam. J. Math. Appl., 3(2) (2020), 185–193. $\href{https://doi.org/10.33401/fujma.790046}{\mbox{[CrossRef]}} $
[5] C. Çiftçi, Disjunctive total domination stability in graphs, Math. Methods Appl. Sci., 44(9) (2021), 7722–7731. $\href{https://doi.org/10.1002/mma.6615 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85087459778?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000545326300001}{\mbox{[Web of Science]}} $
[6] C. Çiftçi and A. Aytaç, Disjunctive total domination in some graphs derived from the subdivision graph, TWMS Journal of Applied and Engineering Mathematics, 11(4) (2021), 1148–1157. $\href{https://www.scopus.com/pages/publications/85119427679?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000704458100019}{\mbox{[Web of Science]}} $
[7] V. Ayta. and T. Turacı, Analysis of vulnerability of some transformation networks, Internat. J. Found. Comput. Sci., 34(01) (2023), 11–24.$ \href{https://doi.org/10.1142/s0129054122500162 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85135165470?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000849387800002}{\mbox{[Web of Science]}} $
[8] C. Çiftçi and V. Aytac¸, Disjunctive total domination in Harary graphs, Comput. J., 66(12) (2023), 2990–2999. $ \href{https://doi.org/10.1093/comjnl/bxac142 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85180528369?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000871579600001}{\mbox{[Web of Science]}} $

[9] T. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, D. Jacobs, J. Knisely and L. Van Der Merwe, Domination subdivision numbers, Discuss. Math. Graph Theory, 21(2) (2001), 239–253.$ \href{https://doi.org/10.7151/dmgt.1147 }{\mbox{[CrossRef]}} $
[10] O. Favaron, H. Karami, R. Khoeilar and S.M. Sheikholeslami, A new bound on total domination subdivision number, Graphs Combin., 25(1) (2009), 41–47. $ \href{https://doi.org/10.1007/s00373-008-0824-6 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/67649639590?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000266335500005}{\mbox{[Web of Science]}} $
[11] T. Haynes, M. Henning and L. Hopkins, Total domination subdivision numbers of graphs, Discuss. Math. Graph Theory, 24(3) (2004), 457–467. $ \href{https://doi.org/10.7151/dmgt.1244 }{\mbox{[CrossRef]}} $
[12] H. Karami, R. Khoeilar, S.M. Sheikholeslami and A. Khodkar, An upper bound for the total domination subdivision number of a graph, Graphs Combin., 25(5) (2009), 727–733. $\href{https://doi.org/10.1007/s00373-010-0877-1 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/77949423697?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000274852300007}{\mbox{[Web of Science]}} $
[13] O. Favaron, H. Karami and S.M. Sheikholeslami, Paired-domination subdivision numbers of graphs, Graphs Combin., 25(4) (2009), 503–512. $\href{https://doi.org/10.1007/s00373-005-0871-1 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/72049094061?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000272669000006}{\mbox{[Web of Science]}} $
[14] A. Babikir, M. Dettlaff, M.A. Henning and M. Lemanska, Independent domination subdivision in graphs, Graphs Combin., 37(3) (2021), 691–709. $ \href{https://doi.org/10.1007/s00373-020-02269-3}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85102699532?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000628458700002}{\mbox{[Web of Science]}}$
[15] M. Atapour, S. Sheikholeslami and A. Khodkar, Roman domination subdivision number of graphs, Aequationes Math., 78(3) (2009), 237– 245. $ \href{https://doi.org/10.1007/s00010-009-2981-y }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/72249095858?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000280299400002}{\mbox{[Web of Science]}} $
[16] K. Haghparast, J. Amjadi, M. Chellali and S.M. Sheikholeslami, On k-Roman domination subdivision number of graphs, AKCE Int. J. Graphs Comb., 19(3) (2022), 261–267. $ \href{https://doi.org/10.1080/09728600.2022.2134836 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85140388136?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000871364800001}{\mbox{[Web of Science]}} $
[17] N. Dehgardi, S. Sheikholeslami and L. Volkmann, The rainbow domination subdivision numbers of graphs, Mat. Vesnik, 67(2) (2015), 102–114. $ \href{https://www.scopus.com/pages/publications/84925940437?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000436976300003}{\mbox{[Web of Science]}} $
[18] C. Çiftçi and V. Aytaç, Disjunctive total domination subdivision number of graphs, Fund. Inform., 174(1) (2020), 15–26. $ \href{https://doi.org/10.3233/fi-2020-1928 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85085242275?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000541024300002}{\mbox{[Web of Science]}} $
[19] C. Çiftçi and A. Aytaç, NP-completeness and bounds for disjunctive total domination subdivision, J. Comb. Optim., 49(1) (2025), 1–10.$\href{https://doi.org/10.1007/s10878-024-01245-4 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85212036356?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001377242700001}{\mbox{[Web of Science]}} $
[20] A. Alex and V. Sangeetha, Secure equitable subdivision number of graphs, Gulf J. Math., 17(2) (2024), 216–225. $ \href{https://doi.org/10.56947/gjom.v17i2.2049 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85206085058?origin=resultslist}{\mbox{[Scopus]}} $
[21] E. Yi, The disjunctive bondage number and the disjunctive total bondage number of graphs, In: Combinatorial Optimization and Applications, Lecture Notes in Computer Science, Springer, Cham, 9486(2015), 660–675. $ \href{https://doi.org/10.1007/978-3-319-26626-8_48 }{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84952033315?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000375220400048}{\mbox{[Web of Science]}} $
[22] S. Intaja and T. Sitthiwarttham, Some graph parameters of fan graph, Int. J. Pure Appl. Math., 80(2) (2012), 217–223. $ \href{https://www.scopus.com/pages/publications/84867826261?origin=resultslist}{\mbox{[Scopus]}} $
[23] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin., 1 (2018), DS6. $\href{https://www.scopus.com/pages/publications/85179355607?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000362654200001}{\mbox{[Web of Science]}} $
[24] A. Aytaç and B. Atay, On domination of a vertex as exponentially, Bull. Int. Math. Virtual Inst., 8 (2018), 271–278. $ \href{https://doi.org/10.7251/BIMVI1802271A}{\mbox{[CrossRef]}} $
[25] I. Javaid and S. Shoukat, On the partition dimension of some wheel related graphs, J. Prime Res. Math., 4(1) (2008), 154–164.$ \href{https://jprm.sms.edu.pk/index.php/jprm/article/view/44}{\mbox{[CrossRef]}} $
[26] K. Koh, D. Rogers, H. Teo and K. Yap, Graceful graphs: some further results and problems, Congr. Numer., 29 (1980), 559–571.

Details

Primary Language

English

Subjects

Applied Mathematics (Other)

Journal Section

Research Article

Authors

Canan Çiftçi ^*
0000-0001-5397-0367
Türkiye

Publication Date

June 30, 2026

Submission Date

January 12, 2026

Acceptance Date

June 29, 2026

Published in Issue

Year 2026 Volume: 9 Number: 2

DOI

https://doi.org/10.33401/fujma.1862064

IZ

https://izlik.org/JA26MD63JN

Cite

RIS / Bibtex

APA

Çiftçi, C. (2026). Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs. Fundamental Journal of Mathematics and Applications, 9(2), 113-122. https://doi.org/10.33401/fujma.1862064

AMA

1.Çiftçi C. Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs. Fundam. J. Math. Appl. 2026;9(2):113-122. doi:10.33401/fujma.1862064

Chicago

Çiftçi, Canan. 2026. “Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs”. Fundamental Journal of Mathematics and Applications 9 (2): 113-22. https://doi.org/10.33401/fujma.1862064.

EndNote

Çiftçi C (June 1, 2026) Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs. Fundamental Journal of Mathematics and Applications 9 2 113–122.

IEEE

[1]C. Çiftçi, “Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs”, Fundam. J. Math. Appl., vol. 9, no. 2, pp. 113–122, June 2026, doi: 10.33401/fujma.1862064.

ISNAD

Çiftçi, Canan. “Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs”. Fundamental Journal of Mathematics and Applications 9/2 (June 1, 2026): 113-122. https://doi.org/10.33401/fujma.1862064.

JAMA

1.Çiftçi C. Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs. Fundam. J. Math. Appl. 2026;9:113–122.

MLA

Çiftçi, Canan. “Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs”. Fundamental Journal of Mathematics and Applications, vol. 9, no. 2, June 2026, pp. 113-22, doi:10.33401/fujma.1862064.

Vancouver

1.Canan Çiftçi. Disjunctive Total Domination Subdivision Number of Cycle and Wheel Related Graphs. Fundam. J. Math. Appl. 2026 Jun. 1;9(2):113-22. doi:10.33401/fujma.1862064