Mathematics and Computational Sciences

Hull numbers of Bicyclic and Tricyclic graphs

Document Type : Original Article

Authors

Kobra Mirzaei
Zahra Yarahmadi

Department of Mathematics, Khor. C., Islamic Azad University, Khorramabad, Iran.

https://doi.org/10.30511/mcs.2026.2090722.1814
Abstract
For two vertices u and v of a connected graph G, the set I[u, v] consists of all vertices lying on some u-v geodesic in G. Let S be a subset of the vertex set V(G), and denote by I[S] the union of all sets I[u, v] with u, v ∈ S. The set S is called convex if I[S] = S. The convex hull [S] of S is defined as the smallest convex set in G containing S. The hull number h(G) of a graph G is the cardinality of a smallest subset S ⊆ V(G) such that [S] = V(G). In this paper, we investigate the hull numbers of bicyclic and tricyclic graphs and provide a classification of these values.

Keywords

Hull number
Bicyclic graphs
Tricyclic graphs
Extreme vertex
Subjects
[1] Buckley, F., Harary, F. (1990). Distance in graphs. New York, NY: Addison–Wesley Longman.
[2] Cáceres, J., Hernando, C., Mora, M., I. M. Pelayo, & Puertas, M. L. (2010). On the geodetic and the hull numbers in strong product graphs. Computers & Mathematics with Applications, 60(11), 3020–3031.
[3] Chartrand, G., Harary, F., & Zhang, P. (2000). On the hull number of a graph. Ars Combinatoria, 57, 129–138.
[4] Chartrand, G., & Zhang, P. (2001). On graphs with a unique minimum hull set. Discussiones Mathematicae Graph Theory, 21(1), 31–42.
[5] Chartrand, G., & Zhang, P. (2001). The forcing hull number of a graph. Journal of Combinatorial Mathematics and Combinatorial Computing, 36, 81–94.
[6] Chartrand, G. (2006). Introduction to graph theory. Tata McGraw–Hill Education.
[7] Chandran, S. V. U., & Parthasarathy, G. J. (2016). The geodesic irredundant sets in graphs. International Journal of Mathematics and Combinatorics, 4, 135–143.
[8] Everett, M. G., & Seidman, S. B. (1985). The hull number of a graph. Discrete Mathematics, 57(3), 217–223.
[9] Yarahmadi, Z. (2020). A classification for bicyclic and tricyclic graphs with respect to Tutte polynomial. Hacettepe Journal of Mathematics and Statistics, 1–11.
Mathematics and Computational Sciences
Volume 7, Issue 3
Summer 2026
Pages 168-178

  • Receive Date 05 June 2026
  • Revise Date 02 July 2026
  • Accept Date 12 July 2026

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