On the flexibility of monophonic paths and connectedness property: application to the human circulatory system

Husniyah Alzubaidi; Ljubiša D. R. Kočinac; Hakeem A. Othman; Husniyah Alzubaidi; Ljubiša D. R. Kočinac; Hakeem A. Othman

doi:10.3934/math.2026773

AIMS Mathematics

2026, Volume 11, Issue 6: 18994-19010. doi: 10.3934/math.2026773

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Research article

On the flexibility of monophonic paths and connectedness property: application to the human circulatory system

1.
Department of Mathematics, AL-Qunfudhah University College, Umm Al-Qura University, KSA
2.
Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia
3.
Department of Mathematics, Albaydha University, Albaydha, Yemen

Received: 24 April 2026 Revised: 01 June 2026 Accepted: 08 June 2026 Published: 29 June 2026
MSC : 05C20, 05C99, 54D05, 54D99

This paper explores the interplay between graph theory and topology by introducing a novel framework based on flexible paths in directed graphs. Extending the classical notion of monophonic paths, we define flexible graphical topologies and characterize classes of directed graphs whose associated flexible graphical topologies are discrete or indiscrete. Fundamental topological properties, including connectedness, compactness, and homeomorphism, are investigated within this framework. Based on these properties, the notions of flexible connectedness and flexible discreteness are introduced and studied. To demonstrate the applicability of the proposed framework, we considered graph representations derived from the human circulatory system. The obtained flexible graphical topological structures are analyzed to illustrate their ability to capture connectedness, flexible connectedness, and flexible discreteness in biologically inspired networks. The results showed that flexible graphical topology provides an extended mathematical framework for studying structural properties of directed networks and offers potential for future applications in biological and complex systems.
- digraph,
- monophonic path,
- topology,
- connectedness,
- continuity
Citation: Husniyah Alzubaidi, Ljubiša D. R. Kočinac, Hakeem A. Othman. On the flexibility of monophonic paths and connectedness property: application to the human circulatory system[J]. AIMS Mathematics, 2026, 11(6): 18994-19010. doi: 10.3934/math.2026773

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Abstract

This paper explores the interplay between graph theory and topology by introducing a novel framework based on flexible paths in directed graphs. Extending the classical notion of monophonic paths, we define flexible graphical topologies and characterize classes of directed graphs whose associated flexible graphical topologies are discrete or indiscrete. Fundamental topological properties, including connectedness, compactness, and homeomorphism, are investigated within this framework. Based on these properties, the notions of flexible connectedness and flexible discreteness are introduced and studied. To demonstrate the applicability of the proposed framework, we considered graph representations derived from the human circulatory system. The obtained flexible graphical topological structures are analyzed to illustrate their ability to capture connectedness, flexible connectedness, and flexible discreteness in biologically inspired networks. The results showed that flexible graphical topology provides an extended mathematical framework for studying structural properties of directed networks and offers potential for future applications in biological and complex systems.

References

[1]	K. A. Abdu, A. Kiliçman, Topologies on the edges set of directed graphs, International Journal of Mathematical Analysis, 12 (2018), 71–84. https://doi.org/10.12988/ijma.2018.814 doi: 10.12988/ijma.2018.814
[2]	R. Abu-Gdairi, A. A. El-Atik, M. K. El-Bably, Topological visualization and graph analysis of rough sets via neighborhoods: A medical application using human heart data, AIMS Mathematics, 8 (2023), 26945–26967. https://doi.org/10.3934/math.20231379 doi: 10.3934/math.20231379
[3]	S. M. J. Amiri, A. Jafarzadeh, H. Khatibzadeh, An Alexandroff topology on graphs, B. Iran. Math. Soc., textbf39 (2013), 647–662.
[4]	Y. Benoist, P. Foulon, F. Labourie, Anosov flows with stable and unstable differentiable distributions, J. Am. Math. Soc., 5 (1992), 33–74. https://doi.org/10.2307/2152750 doi: 10.2307/2152750
[5]	G. Chiaselotti, D. Ciucci, T. Gentile, F. Infusino, Rough set theory and digraphs, Fund. Inform., 153 (2017), 291–325. https://doi.org/10.3233/FI-2017-1542 doi: 10.3233/FI-2017-1542
[6]	M. Dammak, Directed graphs and int-graphic topology, Filomat, 40 (2026), 1893–1903. https://doi.org/10.2298/FIL2605893D doi: 10.2298/FIL2605893D
[7]	F. H. Damag, A. Saif, A. Kiliçman, F. Alhubairah, K. M. Saad, E. E. Ali, et al., Monophonic sets and rough directed topological spaces: Applications with some directed networks, AIMS Mathematics, 10 (2025), 17623–17641. https://doi.org/10.3934/math.2025787 doi: 10.3934/math.2025787
[8]	F. H. Damag, A. Saif, A. Kiliçman, E. E. Ali, M. B. Mesmouli, On $m$-negative sets and out mondirected topologies in the human nervous system, Mathematics, 12 (2024), 3763. https://doi.org/10.3390/math12233763 doi: 10.3390/math12233763
[9]	F. H. Damag, A. Saif, A. Kiliçman, M. B. Mesmouli, F. Alhubairah, Upper a-graphical topological spaces with the COVID-19 form and its diffusion, Axioms, 14 (2025), 84. https://doi.org/10.3390/axioms14020084 doi: 10.3390/axioms14020084
[10]	J. Dugundji, Topology, Boston: Allyn and Bacon, 1966.
[11]	A. E. El-Atik, A. Nawar, M. Atef, Rough approximation models via graphs based on neighborhood systems, Granul. Comput., 6 (2021), 1025–1035. https://doi.org/10.1007/s41066-020-00245-z doi: 10.1007/s41066-020-00245-z
[12]	M. K. El-Bably, R. Abu-Gdairi, K. K. Fleifel, M. A. El-Gayar, Exploring $\beta$-basic rough sets and their applications in medicine, Eur. J. Pure Appl. Math., 17 (2024), 3743–3771. https://doi.org/10.29020/nybg.ejpam.v17i4.5545 doi: 10.29020/nybg.ejpam.v17i4.5545
[13]	M. A. El-Gayar, R. Abu-Gdairi, Extension of topological structures using lattices and rough sets, AIMS Mathematics, 9 (2024), 7552–7569. https://doi.org/10.3934/math.2024366 doi: 10.3934/math.2024366
[14]	A. E. Gomorez, S. R. Canoy, On a topological space generated by monophonic eccentric neighbourhoods of a graph, Eur. J. Pure Appl. Math., 14 (2021), 695–705. https://doi.org/10.29020/nybg.ejpam.v14i3.3990 doi: 10.29020/nybg.ejpam.v14i3.3990
[15]	A. E. Gamorez, C. G. Nianga, S. R. Canoy, Topologies induced by neighbourhoods of a graph under some binary operations, Eur. J. Pure Appl. Math., 12 (2019), 749–755. https://doi.org/10.29020/nybg.ejpam.v12i3.3464 doi: 10.29020/nybg.ejpam.v12i3.3464
[16]	R. A. Hosny, M. K. El-Bably, M. A. El-Gayar, Primal approximation spaces by $\kappa$-neighborhoods with applications, Eur. J. Pure Appl. Math., 18 (2025), 5827. https://doi.org/10.29020/nybg.ejpam.v18i1.5827 doi: 10.29020/nybg.ejpam.v18i1.5827
[17]	R. A. Hosny, N. M. Madbouly, M. K. El-Bably, Theoretical advances and improvements of generalized approximation spaces generation from $\mathbb{I}_{j}$-neighborhoods and ideals with application to Chikungunya disease, Eur. J. Pure Appl. Math., 18 (2025), 6358. https://doi.org/10.29020/nybg.ejpam.v18i4.6358 doi: 10.29020/nybg.ejpam.v18i4.6358
[18]	A. Kiliçman, K. A. Abdu, Topological spaces associated with simple graphs, J. Math. Anal., 9 (2018), 44–52.
[19]	S. Nada, A. El F. El-Atik, M. Atef, New types of topological structures via graphs, Math. Method. Appl. Sci., 41 (2018), 5801–5810. https://doi.org/10.1002/mma.4726 doi: 10.1002/mma.4726
[20]	C. G. S. Nianga, S. R. Canoy, On topologies induced by graphs under some unary and binary operations, Eur. J. Pure Appl. Math., 12 (2019), 499–505. https://doi.org/10.29020/nybg.ejpam.v12i2.3421 doi: 10.29020/nybg.ejpam.v12i2.3421
[21]	C. G. S. Nianga, S. R. Canoy, On a finite topological space induced by Hop neighbourhoods of a graph, Adv. Appl. Discret. Mat., 21 (2019), 79–89. https://doi.org/10.17654/DM021010079 doi: 10.17654/DM021010079
[22]	H. A. Othman, M. M. Al-Shamiri, A. Saif, S. Acharjee, T. Lamoudan, R. Ismail, Pathless directed topology in connection to the circulation of blood in the heart of human body, AIMS Mathematics, 7 (2022), 18158–18172. https://doi.org/10.3934/math.2022999 doi: 10.3934/math.2022999
[23]	H. A. Othman, A. Ayache, A. Saif, On $L_{2}$-directed topological spaces in directed graphs theory, Filomat, 37 (2023), 10005–10013. https://doi.org/10.2298/FIL2329005O doi: 10.2298/FIL2329005O
[24]	H. K. Sarı, A. Kopuzlu, On topological spaces generated by simple undirected graphs, AIMS Mathematics, 5 (2020), 5541–5550. https://doi.org/10.3934/math.2020355 doi: 10.3934/math.2020355
[25]	SARS Expert Committee, SARS in Hong Kong: From Experience to Action, Report of Hong Kong SARS Expert Committee, 2003. Available from: http://www.sars-expertcom.gov.hk/english/reports/reports.html.
[26]	J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, In: Contributions to nonlinear functional analysis New York: Academic Press, 1971, 565–601. http://doi.org/10.1016/B978-0-12-775850-3.50017-0
[27]	M. Shokry, R. E. Aly, Topological properties on graph vs medical application in human heart, International Journal of Applied Mathematics, 15 (2013), 1103–1109.
[28]	F. Syed, S. Khan, M. Toma, Modeling dynamics of the cardiovascular system using fluid-structure interaction methods, Biology, 12 (2023), 1026. https://doi.org/10.3390/biology12071026 doi: 10.3390/biology12071026
[29]	H. O. Zomam, H. A. Othman, M. Dammak, Alexandroff spaces and graphic topology, Advances in Mathematics: Scientific Journal, 10 (2021), 2653–2662. https://doi.org/10.37418/amsj.10.5.28 doi: 10.37418/amsj.10.5.28

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