Mutual-Visibility of Tree and Its Line Graphs

TL;DR

This paper characterizes mutual-visibility sets in trees, introduces the concept of legs, and proves that the mutual-visibility number is preserved under line graph operations, providing bounds for iterated line graphs.

Contribution

It provides a complete characterization of mutual-visibility sets in trees, introduces legs, and establishes the invariance of mutual-visibility number under line graph operations.

Findings

Mutual-visibility sets in trees coincide with leaves of Steiner subtrees.

The mutual-visibility number is preserved under line graph operation for trees.

A lower bound for the mutual-visibility number of iterated line graphs is established.

Abstract

In this paper, we present a complete characterization of mutual-visibility sets in trees. It is shown that a subset $S$ is a mutual-visibility set of a tree $T$ if and only if it coincides with the set of leaves of the Steiner subtree $T\langle S\rangle$. 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. We prove that every tree is absolute-clear. It is further shown that, for every tree $T$ with at least two edges, the mutual-visibility number is preserved under the line graph operation, that is, $\mu(L(T))=\mu(T)$. Examples of unicyclic and block graphs for which this equality fails are also presented. Finally, a tight lower bound for the mutual-visibility number of the iterated line graph is established; namely, $\mu\bigl(L(L(T))\bigr)\ge \left\lfloor \frac{\Delta(T)^2}{3}\right\rfloor$.