Visibility Polynomial of Lollipop, Bistar and Butterfly Graphs

The concept of mutual visibility in graphs provides a useful framework for analyzing how information or influence propagates through networks under structural constraints. In this paper, we study the visibility polynomial for several structured graph classes by exploiting separator properties and distance-based restrictions. After recalling known results for complete graphs, we derive explicit closed-form expressions for the visibility polynomials of lollipop graphs, bistar graphs and butterfly graphs. Our approach makes systematic use of shortest separators and set-separators, which reveal fundamental connections between graph topology and visibility constraints. These results strengthen the theoretical foundations of visibility-based graph invariants and contribute to a deeper understanding of visibility patterns in discrete network models.