Mutual-Visibility Sets in Trees

• Mutual-visibility sets in trees are defined as vertex subsets where the unique path between any two vertices contains no other set member, reflecting maximally dispersed leaf structures.
• They are explicitly characterized by the correspondence with Steiner subtree leaves, leading to the exact formula μ(T) = |leaves| and simplifying analysis in tree topologies.
• Efficient O(|V|+|E|) algorithms compute these sets, demonstrating practical applications in network design, facility placement, and distributed systems.

Mutual-visibility sets in trees constitute a distinct combinatorial invariant with connections to extremal set theory, discrete geometry, and network information flow. For a tree $T=(V,E)$, mutual-visibility sets are intimately linked with structural features such as leaves, branch vertices, and paths, allowing for explicit characterization and efficient enumeration.

1. Definition and Core Concepts

Given a tree $T=(V,E)$, a subset $S\subseteq V$ is a mutual-visibility set if every pair of distinct vertices $u, v\in S$ satisfies that the unique path $P_{uv}$ from $u$ to $v$ in $T$ contains no other member of $S$ as an internal vertex: $$\operatorname{Int}(P_{uv})\cap(S\setminus\{u,v\}) = \emptyset.$$ The mutual-visibility number $\mu(T)$ is the maximum cardinality of any mutual-visibility set in $T$ (Stefano, 2021, Cicerone et al., 2023, B et al., 13 Jan 2026). The term "mutual-visibility" abstractly captures the property that no shortest path between two "visible" points in the set is obstructed by another member of the set.

2. Complete Structural Characterization in Trees

A fundamental structural result is that mutual-visibility sets in trees coincide with the leaf-sets of their own Steiner subtrees. Let $T\langle S\rangle$ denote the minimal subtree containing $S$. Then (B et al., 13 Jan 2026): $$S\ \text{is mutual-visibility in }\ T \iff S = \mathcal{L}(T\langle S\rangle),$$ where $\mathcal{L}(T\langle S\rangle)$ is the set of leaves of the subtree spanned by $S$. This bijective correspondence implies that mutual-visibility sets are maximally dispersed: no two non-leaf points of the same Steiner subtree can both belong to a mutual-visibility set, as one would be internal to the path connecting the others.

3. Explicit Formula for the Mutual-Visibility Number

For any tree $T$,

$\mu(T) = |\mathcal{L}(T)|,$

where $\mathcal{L}(T) = \{v\in V: \deg_T(v) = 1\}$ is the set of leaves (Stefano, 2021, Cicerone et al., 2023, B et al., 13 Jan 2026). This result is exact; no relaxation or approximation is required. Several classic examples follow:

Tree $T$	$\mu(T)$	Specifics
Path $P_n$	2	Leaves are two endpoints
Star $K_{1,n}$	$n$	All leaves but not the center
Full binary tree	$2^h$ (height $h$)	Each leaf is at maximal depth

No non-leaf can ever be included in a maximal mutual-visibility set. Moreover, the formula admits a refinement in terms of the branch (degree ≥ 3) vertices $\Br(T)$: $|\mathcal{L}(T)| = 2 + \sum_{v\in\Br(T)}(\deg_T(v) - 2).$ This follows from the handshaking lemma and holds for every tree (B et al., 13 Jan 2026).

4. Enumeration and Structure of Maximal Sets

When $T$ contains branch vertices, the number of maximal mutual-visibility sets (i.e., sets of size $\mu(T)$) can be computed explicitly via the leg decomposition associated to each leaf. For each leaf $u$, define its leg $L(u)$ as the path from $u$ to its nearest branch vertex $b(u)$, excluding $b(u)$, and let $\ell(u)$ denote its length. The collection of legs is disjoint.

The total number of maximal mutual-visibility sets is then

$r_\mu(T) = \prod_{u\in\mathcal{L}(T)} \ell(u)$

(B et al., 13 Jan 2026). Each such set contains exactly one point on each leg, and no core branch vertex is included. This product formula shows combinatorially how the topology of "dangling paths" amplifies the extremal count.

5. Algorithmic Computation

The structural criteria above lead to an optimal algorithm. For input $T=(V,E)$:

• The maximal mutual-visibility sets are always the set(s) of leaves (one per leg as above).
• Compute the degree of each vertex.
• Collect all vertices of degree 1; these form the unique maximum set.
• Time complexity is $O(|V|+|E|)$, that is, linear in the size of the tree (Stefano, 2021, Cicerone et al., 2023).

This contrasts sharply with the general graph case, where determining $\mu(G)$ is NP-complete (Stefano, 2021).

6. Extensions: Line Graphs, Absolute-Clear Trees, and Glued Trees

Line Graph Preservation

If $T$ is a tree with at least two edges, the mutual-visibility number is preserved in its line graph: $\mu(L(T)) = \mu(T)$ The set of edges incident to leaves corresponds to a maximal mutual-visibility set in $L(T)$ (B et al., 13 Jan 2026).

Absolute-Clear Property

Every tree is absolute-clear: for every nonempty $Q \subseteq V(T)$, maximal absolute $c_Q$-visible sets as defined in (B et al., 13 Jan 2026) are pairwise disjoint.

Glued $t$-ary Trees

For the family of "glued" $t$-ary trees $GT(r, t)$—formed by identifying leaves of two perfect $t$-ary trees—the mutual-visibility number and the enumeration of maximal sets are given by (Roy et al., 2024):

Invariant	Value ($GT(r, t)$)	Number of Extrema
$\mu(GT(r, t))$	$t^r+1$	$2 \times \left(\frac{t^{r+1}-1}{t-1} - t^r\right)$
$\mu_o(GT(r,t))$	$t^r$	1
$\mu_d(GT(r,t))$, $\mu_t(GT(r,t))$	$t^{r-1}(t-1)$	$t^{t^{r-1}}$

This demonstrates the sensitivity of mutual-visibility invariants to global tree structure adjustments.

7. Context, Applications, and Further Results

Mutual-visibility sets serve as extremal subsets where no member obstructs any shortest communication between other members, an abstraction with direct application in network design, facility placement, and distributed systems. In trees, these invariants capture the notion that effectiveness of maximal "seeing" or "coordinating" subsets is always realized at the leaves of minimal Steiner subtrees. The explicit formulas for $\mu(T)$ and the enumeration of maximal sets underline the tractability and algorithmic simplicity available in tree topologies, with preservation under the line graph operation suggesting protocol invariance when moving between node- and edge-centric views in tree-like networks (Stefano, 2021, B et al., 13 Jan 2026, Roy et al., 2024).