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The Total Mutual Visibility Number in Bicyclic Graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 22 شهریور 1405 | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31659.2919 | ||
| نویسندگان | ||
| Golam Hossein Fath-Tabar* ؛ Ensieh Golabi | ||
| University of Kashan | ||
| چکیده | ||
| Let G be a graph. A subset of vertices X ⊆ V (G) is termed a total mutual- visibility set if, for any pair of vertices u, v ∈ V (G), one can find a shortest path connecting them such that none of its internal vertices belong to X. The cardinality of the largest such set is known as the total mutual-visibility number, denoted by μt(G). In this study, we determine this number for the class of connected bicyclic graphs. Our analysis categorizes these graphs into three structural types based on the connection between their two cycles: disjoint cycles linked by a path, cycles intersecting at a single vertex, and cycles sharing a common path (Theta-graphs). For each category, we establish explicit formulas to calculate the exact value of μt(G). These results cover all structural configurations, including variations in cycle lengths, leaf attachments, and internal vertex arrangements. Ultimately, this work provides a complete characterization of the total mutual-visibility number for bicyclic graphs and sets a foundation for future research | ||
| کلیدواژهها | ||
| Bicyclic graphs؛ mutual-visibility set؛ total mutual-visibility set؛ total mutual- visibility number | ||
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