On the Mutual-Visibility of Tree Graphs
Tonny K B, Shikhi M
TL;DR
The paper addresses the problem of characterizing mutual-visibility sets in trees under shortest-path constraints. It introduces a Steiner-subtree based framework and proves that a subset $S$ is mutual-visibility in a tree $T$ if and only if $S$ equals the leaf set of the Steiner subtree $T\langle S\rangle$, yielding $\mu(T)=|\mathcal{L}(T)|$. For trees with branch vertices, it derives a formula for the number of maximal mutual-visibility sets, $r_{\mu}(T)=\prod_{u\in\mathcal{L}(T)} \ell(u)$, in terms of leg lengths, and shows that every tree is absolute-clear, with the mutual-visibility number preserved under line graphs: $\mu(L(T))=\mu(T)$. These results give exact combinatorial characterizations and structural insights, providing a foundation for extending mutual-visibility analysis to broader graph classes.
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, $μ(L(T))=μ(T)$.