Direct Products of Cyclic Semigroups Allowing Apex-Outerplanar Cayley Graphs
DOI:
https://doi.org/10.52575/2687-0959-2026-58-1-44-53Keywords:
Cyclic Semigroups, Direct Products, Cayley Graphs, Apex-Outerplanar Graphs, Outerplanarity, Graph Minors, Graph Embeddings, Commutative Semigroups, Algebraic Graph Theory, Planar Cayley Graphs, Topological Graph Properties, Semigroup Theory, Combinatorial StructuresAbstract
This article examines the structural conditions under which direct products of cyclic semigroups admit Cayley graphs that are apex-outerplanar. A Cayley graph is called apex-outerplanar if the removal of a single distinguished vertex (the apex) yields an outerplanar graph. We provide a characterization of those semigroups whose Cayley graphs possess this property, establishing necessary and sufficient conditions in terms of the generating sets and the algebraic interactions among the cyclic components of the product. The analysis combines techniques from graph theory, including minor theory and embedding arguments, with algebraic properties of commutative semigroups. It is shown that certain configurations of cyclic semigroups give rise to Cayley graphs that become outerplanar upon deletion of an apex vertex, thereby extending known classifications of planar and outerplanar Cayley graphs. These results contribute to the broader understanding of the interplay between algebraic structure and topological properties of Cayley graphs, situating apex-outerplanar graphs within the framework of semigroup theory and flat embeddings.
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References
Dziobiak S, Excluded-minor characterization of apex-outerplanar graphs. LSU Doctoral Dissertations. 3102; 2011. 83 p. https://repository.lsu.edu/gradschool_dissertations/3102
Ding G., Dziobiak S, Excluded-minor characterization of apex-outerplanar graphs. Graphs and Combinatorics. 2016;32(2):583–627.
Chatzidimitriou D. et al., Sparse obstructions for minor-covering parameters. Discrete Appl. Math. 2020;278:28–50.
Donkers H. et al., Preprocessing for outerplanar vertex deletion: an elementary kernel of quartic size. Algorithmica. 2022;84(11):3407–3458.
Eppstein D. On the biplanarity of blowups. J. Graph Algorithms Appl. 2024;28(2):83–99.
Jobson AS., K´ezdy AE, Allminor-minimal apex obstructions with connectivity two. Electron. J. Comb. 2021;28(1):Research Paper P1.23, 58 p.
Leivaditis A. et al., Minor-obstructions for apex sub-unicyclic graphs. Acta Math. Univ. Comen. New Ser. 2019;88(3):903–910.
Leivaditis A. et al., Minor-obstructions for apex sub-unicyclic graphs. Discrete Appl. Math. 2020;284:538–555.
Leivaditis A. et al., Minor obstructions for apex-pseudoforests. Discrete Math. 2021;344(10): Article ID 112529, 31 p.
Mattman TW, Forbidden minors: finding the finite few. in: A primer for undergraduate research. From groups and tiles to frames and vaccines. Cham: Birkh¨auser. 2017;85–97
Sau I. et al., k-apices of minor-closed graph classes. I: Bounding the obstructions. J. Comb. Theory, Ser. B 2023;161:180–227.
Savitsky TJ., Schluchter SA, Some excluded minors for the spindle surface. J. Comb. Math. Comb. Comput. 2024;119:217–232.
Solomatin DV, Direct Products of Cyclic Semigroups Allowing Outerplanar Cayley Graphs and Their Generalizations. Applied Mathematics & Physics. 2024;56(1):13–20. (in Russ.)
Harary F. Graph Theory: Advanced Book Program Series. Boulder: Westview Press; 1994. 284 p.
Zelinka B. Graphs of Semigroups. Casopis. Pest. Mat. 1981;106:407–408.
Solomatin DV, Researches of semigroups with planar Cayley graphs: results and problems. Prikladnaya Diskretnaya Matematika. 2021;54:5–57. (in Russ.)
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