Visibility Polynomial in Graph Theory

• Visibility Polynomial is a generating function that counts mutual-visibility sets by size, capturing the full distribution beyond the maximum mutual-visibility number.
• The polynomial refines graph invariants through explicit coefficient formulas, hereditary constraints, and inclusion–exclusion principles in diverse graph families and operations.
• It extends to variants such as dual, outer, and total visibility polynomials, linking mutual visibility with general position sets and influencing graph optimization.

In graph theory, the visibility polynomial is a generating function that counts mutual-visibility sets by cardinality. For a simple graph $G$, if $r_i$ denotes the number of mutual-visibility sets of size $i$, then the visibility polynomial is written as $V(G)=\sum_{i\ge 0} r_i x^i$ in one line of work, while later papers also use $\nu(G)$ and $\mathcal{V}(G)$ for the same counting invariant. The concept was introduced explicitly for graphs in "Visibility polynomials, dual visibility spectrum, and characterization of total mutual-visibility sets" (Bujtás et al., 2024), and subsequently developed for concrete graph classes, joins, and corona products (B et al., 2 Jul 2025, B et al., 26 Sep 2025, B et al., 2 Sep 2025). Earlier visibility literature in Delaunay meshes and in graph optimization studied visibility relations and visibility numbers, but did not define a visibility polynomial (Peters, 2015, Bilò et al., 2024).

1. Foundational definition

Let $G=(V,E)$ be a simple graph and let $X\subseteq V$. Two vertices $u,v\in V$ are $X$-visible if there exists a shortest $r_i$0-path $r_i$1 such that

$r_i$2

A set $r_i$3 is a mutual-visibility set if every pair of vertices in $r_i$4 is $r_i$5-visible. Its maximum cardinality is the mutual-visibility number $r_i$6 (Bujtás et al., 2024).

The visibility polynomial packages the full counting sequence of such sets: $r_i$7 where $r_i$8 is the number of mutual-visibility sets of cardinality $r_i$9. The constant term is always $i$0, because the empty set is a mutual-visibility set, and the degree is $i$1 (Bujtás et al., 2024). In the notation of later papers,

$i$2

counts the same object (B et al., 2 Jul 2025, B et al., 26 Sep 2025).

A basic universal fact is that every set of size $i$3, $i$4, or $i$5 is automatically a mutual-visibility set. Hence, for a graph on $i$6 vertices, the polynomial begins

$i$7

(B et al., 2 Jul 2025). The polynomial is an isomorphism invariant, but not a complete one: non-isomorphic graphs may share the same visibility polynomial; one explicit example has

$i$8

(B et al., 2 Jul 2025).

This counting viewpoint refines $i$9. The number $V(G)=\sum_{i\ge 0} r_i x^i$0 records only the largest feasible size, while the polynomial records the entire distribution of mutual-visibility sets across all sizes. This suggests why later work treats it as a structural invariant rather than merely an optimization parameter.

2. Variants and coefficient structure

The 2024 formulation introduced parallel set systems and parallel polynomials for three stronger visibility notions (Bujtás et al., 2024). For $V(G)=\sum_{i\ge 0} r_i x^i$1:

• $V(G)=\sum_{i\ge 0} r_i x^i$2 is an outer mutual-visibility set if every two vertices $V(G)=\sum_{i\ge 0} r_i x^i$3 are $V(G)=\sum_{i\ge 0} r_i x^i$4-visible, and every $V(G)=\sum_{i\ge 0} r_i x^i$5, $V(G)=\sum_{i\ge 0} r_i x^i$6 are $V(G)=\sum_{i\ge 0} r_i x^i$7-visible.
• $V(G)=\sum_{i\ge 0} r_i x^i$8 is a dual mutual-visibility set if every two vertices of $V(G)=\sum_{i\ge 0} r_i x^i$9 and every two vertices of $\nu(G)$0 are $\nu(G)$1-visible.
• $\nu(G)$2 is a total mutual-visibility set if any two vertices of $\nu(G)$3 are $\nu(G)$4-visible.

The corresponding maxima are denoted

$\nu(G)$5

and the associated generating functions are the visibility polynomial $\nu(G)$6, the dual visibility polynomial $\nu(G)$7, the outer visibility polynomial $\nu(G)$8, and the total visibility polynomial $\nu(G)$9 (Bujtás et al., 2024).

A central structural asymmetry is that dual mutual-visibility sets are not closed under taking subsets. The paper gives the example of two adjacent vertices in $\mathcal{V}(G)$0: together they form a dual mutual-visibility set, but neither singleton does (Bujtás et al., 2024). By contrast, ordinary, outer, and total mutual-visibility families behave downward-closed, and this hereditary behavior imposes coefficient constraints.

One such constraint is a Kruskal–Katona type lower-shadow inequality. If $\mathcal{V}(G)$1 for some real $\mathcal{V}(G)$2, then for $\mathcal{V}(G)$3,

$\mathcal{V}(G)$4

(Bujtás et al., 2024). The same hereditary structure yields an inclusion–exclusion formula over maximal sets: if $\mathcal{V}(G)$5 are the maximal mutual-visibility sets of $\mathcal{V}(G)$6, then

$\mathcal{V}(G)$7

where $\mathcal{V}(G)$8 and $\mathcal{V}(G)$9 (Bujtás et al., 2024).

The dual case behaves in a radically different manner. The paper formalizes its coefficient sequence as the dual visibility spectrum

$G=(V,E)$0

where $G=(V,E)$1 and $G=(V,E)$2 counts dual mutual-visibility sets of size $G=(V,E)$3. Its realization theorem states that for every $G=(V,E)$4 and every sequence of nonnegative integers

$G=(V,E)$5

there exists a graph $G=(V,E)$6 with $G=(V,E)$7 whose dual visibility spectrum is exactly

$G=(V,E)$8

(Bujtás et al., 2024). Consequently, every polynomial with nonnegative integer coefficients and constant term $G=(V,E)$9 is the dual visibility polynomial of some graph (Bujtás et al., 2024). This establishes a sharp dichotomy: ordinary, outer, and total visibility polynomials are constrained by hereditary combinatorics, whereas dual visibility polynomials are universal.

3. Relations to general position and total visibility

The visibility polynomial is closely related to the general position polynomial on geodetic graphs. A set $X\subseteq V$0 is a general position set if no three distinct vertices of $X\subseteq V$1 lie on a common shortest path. In a geodetic graph, where there is a unique shortest path between every pair of vertices, mutual-visibility sets and general position sets coincide: $X\subseteq V$2 is a mutual-visibility set if and only if $X\subseteq V$3 is a general position set (Bujtás et al., 2024). As a result, on geodetic graphs the visibility polynomial is exactly the general position polynomial.

The Petersen graph provides a concrete example. It is geodetic, and the paper proves that a set $X\subseteq V$4 is an outer mutual-visibility set if and only if $X\subseteq V$5 is an independent set. It then computes

$X\subseteq V$6

and

$X\subseteq V$7

Since $X\subseteq V$8,

$X\subseteq V$9

(Bujtás et al., 2024).

The total variant also admits a clean structural characterization. For a connected graph $u,v\in V$0 and $u,v\in V$1, the following are equivalent:

1. $u,v\in V$2 is a total mutual-visibility set.
2. Every pair of vertices $u,v\in V$3 with $u,v\in V$4 is $u,v\in V$5-visible.
3. For every pair $u,v\in V$6 with $u,v\in V$7,

$u,v\in V$8

equivalently,

$u,v\in V$9

This reduces total mutual-visibility testing to distance-$X$0 neighborhoods (Bujtás et al., 2024). In geodetic graphs, total mutual-visibility is governed by simplicial vertices: $X$1 where $X$2 is the number of simplicial vertices, and the set of all simplicial vertices is the unique $X$3-set (Bujtás et al., 2024).

A related but distinct comparison concerns the mutual-visibility number and the general position number $X$4. On graphs of diameter $X$5, the ratio

$X$6

can be as large as

$X$7

and there is also the upper bound

$X$8

(Bilò et al., 2024). This suggests that the visibility polynomial and the general position polynomial may diverge sharply outside the geodetic regime.

4. Explicit formulas for graph classes

A substantial part of the literature derives closed forms for families with rigid geodesic structure. The following ordinary visibility polynomials are stated explicitly.

Graph family	Visibility polynomial	Source
$X$9	$r_i$00	(B et al., 2 Jul 2025)
$r_i$01	$r_i$02	(B et al., 2 Jul 2025)
$r_i$03	$r_i$04	(B et al., 2 Jul 2025)
$r_i$05	$r_i$06	(B et al., 26 Sep 2025)
$r_i$07	$r_i$08	(B et al., 26 Sep 2025)
$r_i$09	$r_i$10	(B et al., 26 Sep 2025)

For cycles, the nontrivial coefficient is $r_i$11, since $r_i$12. The exact formula is

$r_i$13

The same paper proves that the number of maximal mutual-visibility sets satisfies

$r_i$14

and

$r_i$15

(B et al., 2 Jul 2025).

Complete bipartite graphs admit explicit coefficient formulas as well. For $r_i$16, assuming $r_i$17 and $r_i$18,

$r_i$19

with piecewise formulas for $r_i$20, and in particular $r_i$21 (B et al., 2 Jul 2025). For the balanced case $r_i$22 with $r_i$23, the 2024 paper gives the parallel family

$r_i$24

$r_i$25

$r_i$26

(Bujtás et al., 2024).

The same 2024 paper records simple variant formulas for paths $r_i$27: $r_i$28 and for $r_i$29,

$r_i$30

(Bujtás et al., 2024).

These formulas display a recurring pattern. Many families decompose into a large term such as $r_i$31, representing unrestricted subsets of a geodesically “safe” subgraph, plus low-degree correction terms arising from a distinguished hub, apex, or separator. This suggests that visibility polynomials are especially tractable on graphs whose shortest-path structure is dominated by a small number of central vertices.

5. Graph operations and computation

Beyond individual families, the literature develops formulas under standard graph operations. For disconnected graphs with connected components $r_i$32,

$r_i$33

since every mutual-visibility set lies inside a single component (B et al., 2 Jul 2025).

For joins, the situation is controlled by the fact that

$r_i$34

If $r_i$35 and $r_i$36 are disjoint complete graphs, then

$r_i$37

recovering the identity $r_i$38 for $r_i$39 (B et al., 2 Jul 2025). For general joins of non-complete graphs, the coefficients admit an explicit piecewise description in terms of clique counts $r_i$40 and counts $r_i$41 of mutual-visibility sets of size $r_i$42 and diameter $r_i$43 (B et al., 2 Jul 2025).

The corona product $r_i$44 yields an even more rigid visibility structure. If both $r_i$45 and $r_i$46 have at least two vertices, then

$r_i$47

and the maximum mutual-visibility set is exactly the union of all attached copies: $r_i$48 Hence $r_i$49 is monic, because there is exactly one mutual-visibility set of maximum size (B et al., 2 Sep 2025). The main formula is

$r_i$50

where $r_i$51, $r_i$52, and the $r_i$53 terms are governed by $r_i$54-visible sets and inclusion–exclusion (B et al., 2 Sep 2025).

On the algorithmic side, the direct computation problem is expensive. A known mutual-visibility test on $r_i$55 runs in

$r_i$56

and exhaustive enumeration over all subsets yields total time

$r_i$57

summarized as essentially $r_i$58 in the worst case (B et al., 2 Jul 2025). This computational cost explains the emphasis on structural formulas for graph classes and graph products (B et al., 26 Sep 2025).

6. Scope, terminology, and neighboring uses of “visibility”

The graph-theoretic visibility polynomial should be distinguished from several adjacent notions in the visibility literature. In "Visibility in Proximal Delaunay Meshes" (Peters, 2015), visibility is formulated as a relation $r_i$59 on subsets of a Delaunay mesh, with ordinary visibility corresponding to a common triangle vertex and strong visibility to a common edge. The main result is that the visibility relation is equivalent to Wallman proximity, and that a Delaunay triangulation region endowed with $r_i$60 has a local Leader uniform topology. That paper does not define a visibility polynomial (Peters, 2015).

Similarly, "On the approximability of graph visibility problems" (Bilò et al., 2024) studies four optimization invariants—mutual, outer mutual, dual mutual, and total mutual visibility—and gives approximation and inapproximability results. It does not define a polynomial, generating function, or algebraic enumerator of visibility sets (Bilò et al., 2024).

A second source of ambiguity arises in lattice-point visibility. In "Visibility of Lattice Points across Polynomials" (Ahuja, 22 Jan 2026), one studies visibility along the polynomial family

$r_i$61

with gcd-based and lcm-based criteria for when a lattice point is visible along such a curve. In "Lattice point visibility along powers of polynomials" (Lobzenz et al., 24 Apr 2026), visibility density is analyzed for polynomial lines of sight, and the Visibility Density Conjecture is proved for a large class of polynomials. In this literature, the word polynomial modifies the geometric line of sight rather than denoting a graph-counting invariant (Ahuja, 22 Jan 2026, Lobzenz et al., 24 Apr 2026).

The established usage in graph theory is therefore precise: a visibility polynomial is a generating function enumerating mutual-visibility sets, together with its outer, dual, and total variants. Outside graph theory, “visibility” and “polynomial” can co-occur in different senses, but those senses refer to different objects.