Visibility polynomial of corona of two graphs

Abstract

In multiagent systems, effective coordination, coverage, and communication often rely on the concept of visibility between agents or nodes within the system. Graph-theoretically, for any subset $X$ of vertices of a graph $G$, two vertices are said to be $X$-visible if there exists a shortest path between them that contains no vertex of $X$ as an internal vertex. In this paper, we investigate the visibility polynomial associated with the corona product of two graphs. The visibility polynomial encodes the number of mutual-visibility sets of all orders within a graph, and the process of enumerating these sets provides a deeper understanding of their structural properties. We characterize the structure of mutual-visibility sets arising specifically within the corona product. As part of this study, we introduce the notion of $C_Q$-visible sets, defined with respect to a selected subset $Q$ of vertices in a graph $G$. A $C_Q$-visible set is a collection of vertices in $\overline{Q}$ that is not only $Q$-visible, but also individually visible from each vertex in $Q$. Using this concept, we establish several characterizations and properties of mutual-visibility sets within the corona product, thereby providing deeper insights into their structure and behavior.