On the Mutual-Visibility of Tree Graphs

Abstract: The notion of mutual visibility in graphs arises from constraining shortest paths by forbidding internal vertices from belonging to a specified subset. Mutual-visibility sets, originally introduced as a tool for studying information flow and structural restrictions in complex networks, have since gained increasing attention due to their theoretical significance and diverse applications. In this paper, a complete characterization of mutual-visibility sets in trees is presented. It is shown that a subset $S$ is a mutual-visibility set of $T$ if and only if it coincides with the set of leaves of the Steiner subtree $T\langle S\rangle$. As a consequence, the mutual-visibility number of a tree is equal to the number of its leaves. For trees containing branch vertices, the notion of legs is introduced and an explicit formula for the number of maximal mutual-visibility sets is derived in terms of the corresponding leg lengths. It is proved that every tree is absolute-clear. It is further established that the mutual-visibility number is preserved under the line graph operation for trees with at least two edges, that is, $μ(L(T))=μ(T)$.