Mutual-Visibility of Tree and Its Line Graphs

Shikhi M; Tonny K B

arxiv: 2601.08270 · v2 · pith:574VZBH6new · submitted 2026-01-13 · 🧮 math.CO

Mutual-Visibility of Tree and Its Line Graphs

Tonny K B , Shikhi M This is my paper

Pith reviewed 2026-05-21 16:44 UTC · model grok-4.3

classification 🧮 math.CO

keywords mutual-visibilitySteiner subtreeline graphstreesvisibility numbermaximal setslegsiterated line graphs

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The pith

In a tree, a vertex set allows mutual visibility exactly when it matches the leaves of the minimal subtree connecting them.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives a complete characterization of which subsets of vertices in a tree are mutual-visibility sets. A set S qualifies if and only if it is precisely the leaves of the Steiner subtree formed by S. This reduces the visibility condition to a structural property of the connecting subtree. The work also shows that this mutual-visibility number remains unchanged when the tree is replaced by its line graph, provided the tree has at least two edges. This preservation result extends to a lower bound on the number for the second iterated line graph.

Core claim

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⟨S⟩. For every tree T with at least two edges, the mutual-visibility number satisfies μ(L(T)) = μ(T). Every tree is absolute-clear, and an explicit formula for the number of maximal mutual-visibility sets is given using leg lengths when branch vertices are present.

What carries the argument

The Steiner subtree T⟨S⟩, the unique minimal subtree connecting the vertices in S; it identifies mutual-visibility sets as exactly those S that are the leaves of this subtree.

If this is right

The number of maximal mutual-visibility sets in a tree with branch vertices can be calculated directly from the lengths of its legs.
Every tree satisfies the absolute-clear property with respect to mutual visibility.
The mutual-visibility number is invariant under taking the line graph for trees with two or more edges.
A lower bound of floor of delta squared over 3 holds for the mutual-visibility number of the second iterated line graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the characterization holds, algorithms for finding large mutual-visibility sets in trees could focus on selecting leaf sets of candidate subtrees.
The equality with line graphs may indicate that visibility numbers are stable under certain graph transformations that preserve path structures.
Similar Steiner-subtree reductions might apply to mutual visibility in other acyclic graphs or distance-based properties.

Load-bearing premise

The proofs depend on trees having a unique Steiner subtree for every vertex set and on the mutual-visibility definition requiring that shortest paths between set members have no other set members as internal vertices.

What would settle it

Finding a tree and a subset S that is mutually visible but not equal to the leaves of T⟨S⟩, or a tree T with at least two edges where the mutual-visibility number of its line graph differs from that of T.

read the original 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, $\mu\bigl(L(L(T))\bigr)\ge \left\lfloor \frac{\Delta(T)^2}{3}\right\rfloor$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a direct characterization of mutual-visibility sets in trees via Steiner subtree leaves plus an invariance result under line graphs, but the core equivalence is a straightforward consequence of unique paths.

read the letter

The main takeaway is that a set S forms a mutual-visibility set in a tree T exactly when it matches the leaves of the Steiner subtree T⟨S⟩, and that μ(L(T)) equals μ(T) for any tree with at least two edges. They also supply a counting formula for maximal sets using leg lengths on branch vertices, prove every tree is absolute-clear, give counterexamples where the equality fails for unicyclic and block graphs, and state a lower bound of floor(Δ(T)^2 / 3) for the double line graph. The characterization and the invariance look like the real contributions here. The leg-length formula adds a usable explicit count that prior visibility work on trees apparently lacked, and the counterexamples help bound where the invariance stops. Those pieces are concrete enough to be checked against small trees. The lower bound is presented as tight, which is worth verifying in examples but at least gives a concrete number to work with. The central equivalence follows pretty directly once you recall that trees have unique paths, so any internal vertex on a path between two points in S would violate the visibility condition unless S sits exactly at the leaves of the minimal connecting subtree. That part is standard tree structure rather than a deep new insight, though spelling it out cleanly for this parameter is still useful. I don't see circularity or invented parameters that break the logic, and the claims rest on ordinary Steiner subtree properties. One minor soft spot is that the absolute-clear property is asserted without much extra payoff beyond the main characterization, and edge cases like paths or stars with exactly two edges could use explicit checks in the write-up. The proofs aren't visible here, but the stress-test note suggests no load-bearing gaps in the reductions to line graphs. This work sits squarely in combinatorial graph theory on visibility numbers. A reader already following papers on mutual-visibility or similar parameters on trees and their line graphs would pick up the formulas and the invariance without much trouble. It has enough specific results and counterexamples to merit a serious referee rather than a desk reject, mainly to confirm the counting formula and the tightness claim hold in full generality. I would send it out for review.

Referee Report

0 major / 3 minor

Summary. The manuscript characterizes mutual-visibility sets in trees: 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⟨S⟩. For trees with branch vertices it introduces legs and derives an explicit formula for the number of maximal mutual-visibility sets in terms of leg lengths. It proves every tree is absolute-clear, shows that μ(L(T)) = μ(T) for every tree T with at least two edges, supplies counter-examples for unicyclic and block graphs, and establishes the tight lower bound μ(L(L(T))) ≥ ⌊Δ(T)²/3⌋.

Significance. If the central claims hold, the work supplies a clean structural characterization of mutual-visibility sets that follows directly from the unique-path property of trees. The invariance result under the line-graph operation and the explicit lower bound on the second iterated line graph are useful for understanding the behavior of the parameter under standard graph operations. The leg-based counting formula provides a concrete combinatorial tool. These features, together with the absence of free parameters or fitted quantities, strengthen the contribution to visibility parameters in graph theory.

minor comments (3)

[Abstract] The abstract introduces the term 'absolute-clear' without a definition or forward reference; this should be defined in the preliminaries or at first use in the introduction.
[Section introducing legs and the counting formula] The leg-length counting formula is stated in terms of leg lengths, but the derivation would benefit from an explicit small example (e.g., a tree with three legs of lengths 2, 3, 1) to illustrate how the formula is obtained from the leaf-set characterization.
[Proof of the line-graph invariance] The proof that μ(L(T)) = μ(T) for trees with exactly two edges should be written out separately rather than subsumed under the general case, to address the edge-case concern raised in the scope statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main results, including the characterization via Steiner subtrees, the leg-based counting formula, the absolute-clear property, the invariance μ(L(T)) = μ(T), and the lower bound on μ(L(L(T))). We appreciate the recognition of the structural and combinatorial contributions.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central characterization—that a set S is a mutual-visibility set of tree T iff S equals the leaves of the Steiner subtree T⟨S⟩—is derived directly from the unique-path property of trees and the given definition of mutual visibility (shortest paths between pairs in S have no internal vertices from S). This equivalence is proven from first principles without fitting parameters or self-referential equations. The result μ(L(T)) = μ(T) follows by applying the same leaf-set characterization to the block structure of the line graph, again without reducing any derived quantity to a quantity defined in terms of itself. No self-citations are load-bearing, no ansatzes are smuggled, and no uniqueness theorems are imported from prior author work. The derivation chain is self-contained against standard graph-theoretic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard definition of Steiner subtrees in trees, the definition of mutual visibility, and the combinatorial properties of line graphs. No numerical parameters are fitted. The notion of legs is introduced specifically for the counting result.

axioms (2)

standard math Every finite set of vertices in a tree induces a unique Steiner subtree.
Invoked implicitly when the paper equates mutual-visibility sets with the leaves of T⟨S⟩.
domain assumption The line graph L(T) of a tree T with at least two edges has the same mutual-visibility number as T.
This is the statement being proved rather than an axiom, but the proof must rely on the structural correspondence between vertices of T and edges of L(T).

invented entities (1)

legs no independent evidence
purpose: To parameterize the counting formula for the number of maximal mutual-visibility sets in trees that contain branch vertices.
Introduced in the abstract as a new auxiliary notion whose lengths determine the count.

pith-pipeline@v0.9.0 · 5704 in / 1540 out tokens · 42830 ms · 2026-05-21T16:44:23.148598+00:00 · methodology

Mutual-Visibility of Tree and Its Line Graphs

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)