---
title: Visibility Polynomial in Graph Theory
url: https://www.emergentmind.com/topics/visibility-polynomial
type: topic
---

# Visibility Polynomial in Graph Theory

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" [2412.03066], and subsequently developed for concrete graph classes, joins, and corona products [2507.01851], [2509.22571], [2509.02509]. Earlier visibility literature in Delaunay meshes and in graph optimization studied visibility relations and visibility numbers, but did not define a visibility polynomial [1501.02357], [2407.00409].

## 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 \(u,v\)-path \(P\) such that
\[
V(P)\cap X \subseteq \{u,v\}.
\]
A set \(X\subseteq V\) is a **mutual-visibility set** if every pair of vertices in \(X\) is \(X\)-visible. Its maximum cardinality is the **mutual-visibility number** \(\mu(G)\) [2412.03066].

The visibility polynomial packages the full counting sequence of such sets:
\[
V(G)=\sum_{i\ge 0} r_i x^i,
\]
where \(r_i\) is the number of mutual-visibility sets of cardinality \(i\). The constant term is always \(1\), because the empty set is a mutual-visibility set, and the degree is \(\mu(G)\) [2412.03066]. In the notation of later papers,
\[
\nu(G)=\sum_{i\ge 0} r_i x^i
\quad\text{or}\quad
\mathcal{V}(G)=\sum_{k\ge 0} v_k x^k
\]
counts the same object [2507.01851], [2509.22571].

A basic universal fact is that every set of size \(0\), \(1\), or \(2\) is automatically a mutual-visibility set. Hence, for a graph on \(n\) vertices, the polynomial begins
\[
1+nx+\binom{n}{2}x^2+\cdots
\]
[2507.01851]. The polynomial is an **isomorphism invariant**, but not a complete one: non-isomorphic graphs may share the same visibility polynomial; one explicit example has
\[
\mathcal{V}(G_1)=\mathcal{V}(G_2)=1+4x+6x^2+4x^3
\]
[2507.01851].

This counting viewpoint refines \(\mu(G)\). The number \(\mu(G)\) 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 [2412.03066]. For \(X\subseteq V(G)\):

- \(X\) is an **outer mutual-visibility set** if every two vertices \(x,y\in X\) are \(X\)-visible, and every \(x\in X\), \(y\notin X\) are \(X\)-visible.
- \(X\) is a **dual mutual-visibility set** if every two vertices of \(X\) and every two vertices of \(V(G)\setminus X\) are \(X\)-visible.
- \(X\) is a **total mutual-visibility set** if any two vertices of \(V(G)\) are \(X\)-visible.

The corresponding maxima are denoted
\[
\mu(G),\qquad \mu_t(G),\qquad \mu_d(G),\qquad \mu_o(G),
\]
and the associated generating functions are the **visibility polynomial** \(V(G)\), the **dual visibility polynomial** \(V_d(G)\), the **outer visibility polynomial** \(V_o(G)\), and the **total visibility polynomial** \(V_t(G)\) [2412.03066].

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 \(C_6\): together they form a dual mutual-visibility set, but neither singleton does [2412.03066]. 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 \(r_i=\binom{z}{i}\) for some real \(z\), then for \(P\in\{V,V_o,V_t\}\),
\[
r_{i-1}\ge \binom{z}{i-1}
\]
[2412.03066]. The same hereditary structure yields an inclusion–exclusion formula over maximal sets: if \(X_1,\dots,X_n\) are the maximal mutual-visibility sets of \(G\), then
\[
P(G)=\sum_{k=1}^{n}(-1)^{k-1}\!\!\!\sum_{\{i_1,\dots,i_k\}\subseteq [n]}
P(X_{i_1}\cap\cdots\cap X_{i_k}),
\]
where \(P(X)=(1+x)^{|X|}\) and \(P\in\{V,V_o,V_t\}\) [2412.03066].

The dual case behaves in a radically different manner. The paper formalizes its coefficient sequence as the **dual visibility spectrum**
\[
(r_0,r_1,\dots,r_k),
\]
where \(k=\mu_d(G)\) and \(r_i\) counts dual mutual-visibility sets of size \(i\). Its realization theorem states that for every \(k>0\) and every sequence of nonnegative integers
\[
r_0=1,\ r_1,\dots,r_k \quad\text{with } r_k>0,
\]
there exists a graph \(G\) with \(\mu_d(G)=k\) whose dual visibility spectrum is exactly
\[
(1,r_1,\dots,r_k)
\]
[2412.03066]. Consequently, **every polynomial with nonnegative integer coefficients and constant term \(1\) is the dual visibility polynomial of some graph** [2412.03066]. 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 \(X\subseteq V(G)\) is a general position set if no three distinct vertices of \(X\) 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: \(X\) is a mutual-visibility set if and only if \(X\) is a general position set [2412.03066]. 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 \(X\subseteq V(P)\) is an outer mutual-visibility set if and only if \(X\) is an independent set. It then computes
\[
V_o(P)=1+10x+30x^2+30x^3+5x^4,
\]
and
\[
V(P)=1+10x+45x^2+90x^3+80x^4+30x^5+5x^6.
\]
Since \(\mu_d(P)=\mu_t(P)=0\),
\[
V_d(P)=V_t(P)=1
\]
[2412.03066].

The total variant also admits a clean structural characterization. For a connected graph \(G\) and \(X\subseteq V(G)\), the following are equivalent:

1. \(X\) is a total mutual-visibility set.
2. Every pair of vertices \(u,v\) with \(d_G(u,v)=2\) is \(X\)-visible.
3. For every pair \(u,v\) with \(d_G(u,v)=2\),
   \[
   N_G(u)\cap N_G(v)\not\subseteq X,
   \]
   equivalently,
   \[
   N_G(u)\cap N_G(v)\setminus X\neq\emptyset.
   \]

This reduces total mutual-visibility testing to distance-\(2\) neighborhoods [2412.03066]. In geodetic graphs, total mutual-visibility is governed by simplicial vertices:
\[
\mu_t(G)=s(G),
\]
where \(s(G)\) is the number of simplicial vertices, and the set of all simplicial vertices is the unique \(\mu_t\)-set [2412.03066].

A related but distinct comparison concerns the mutual-visibility number and the general position number \(gp(G)\). On graphs of diameter \(2\), the ratio
\[
\frac{\mu(G)}{gp(G)}
\]
can be as large as
\[
\Omega\left(\frac{n}{\log^2 n}\right),
\]
and there is also the upper bound
\[
\mu(G)\le O\left(\frac{n}{\log n}\right)\, gp(G)
\]
[2407.00409]. 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 |
|---|---|---|
| \(K_n\) | \((1+x)^n\) | [2507.01851] |
| \(P_n\) | \(1+nx+\binom{n}{2}x^2\) | [2507.01851] |
| \(C_n\) | \(1+nx+\binom{n}{2}x^2+r_3x^3\) | [2507.01851] |
| \(W_n,\ n\ge 8\) | \((1+x)^{n-1}+x+(n-1)x^2+2(n-1)x^3\) | [2509.22571] |
| \(F_n\) | \((1+x)^{2n}+x+2n x^2+n x^3\) | [2509.22571] |
| \(S_n,\ n\ge 3\) | \((1+x)^{n-1}+x+(n-1)x^2+(2n-5)x^3\) | [2509.22571] |

For cycles, the nontrivial coefficient is \(r_3\), since \(\mu(C_n)=3\). The exact formula is
\[
r_3=
\begin{cases}
\dfrac{n(n^2-1)}{24}, & \text{if } n \text{ is odd},\\[6pt]
\dfrac{(n-2)n(n+8)}{24}, & \text{if } n \text{ is even}.
\end{cases}
\]
The same paper proves that the number of maximal mutual-visibility sets satisfies
\[
r_\mu(C_{n_1})=r_\mu(C_{n_2})
\quad \Longleftrightarrow \quad
\{n_1,n_2\}=\{6,7\},
\]
and
\[
r_\mu(C_6)=r_\mu(C_7)=14
\]
[2507.01851].

Complete bipartite graphs admit explicit coefficient formulas as well. For \(K_{m,n}\), assuming \(m,n\ge 3\) and \(m\le n\),
\[
\mathcal{V}(K_{m,n})=\sum_{i=0}^{m+n-2} r_i x^i,
\]
with piecewise formulas for \(r_i\), and in particular \(\mu(K_{m,n})=m+n-2\) [2507.01851]. For the balanced case \(K_{n,n}\) with \(n\ge 3\), the 2024 paper gives the parallel family
\[
V(K_{n,n})=\big((x+1)^n-x^n\big)^2+2nx^n+1+2x^{2n},
\]
\[
V_o(K_{n,n})=\big((x+1)^n-x^n\big)^2+2x^{2n},
\]
\[
V_d(K_{n,n})=V_t(K_{n,n})=\big((x+1)^n-x^n\big)^2
\]
[2412.03066].

The same 2024 paper records simple variant formulas for paths \(P_n\):
\[
V(P_n)=1+nx+\cdots,
\]
and for \(n\ge 3\),
\[
V_d(P_n)=1+2x+3x^2,\qquad
V_o(P_n)=1+nx+x^2,\qquad
V_t(P_n)=1+2x+x^2
\]
[2412.03066].

These formulas display a recurring pattern. Many families decompose into a large term such as \((1+x)^k\), 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 \(G_1,\dots,G_m\),
\[
\nu(G)=\nu(G_1)+\nu(G_2)+\cdots+\nu(G_m)-m+1,
\]
since every mutual-visibility set lies inside a single component [2507.01851].

For joins, the situation is controlled by the fact that
\[
\operatorname{diam}(G\vee H)=2.
\]
If \(G\) and \(H\) are disjoint complete graphs, then
\[
\mathcal{V}(G\vee H)=\mathcal{V}(G)\mathcal{V}(H),
\]
recovering the identity \((1+x)^{m+n}=(1+x)^m(1+x)^n\) for \(K_m\vee K_n\cong K_{m+n}\) [2507.01851]. For general joins of non-complete graphs, the coefficients admit an explicit piecewise description in terms of clique counts \(c_k(G)\) and counts \(\mu_{k,2}(G)\) of mutual-visibility sets of size \(k\) and diameter \(2\) [2507.01851].

The corona product \(G\odot H\) yields an even more rigid visibility structure. If both \(G\) and \(H\) have at least two vertices, then
\[
\mu(G\odot H)=|V(G)|\,|V(H)|,
\]
and the maximum mutual-visibility set is exactly the union of all attached copies:
\[
M=\bigcup_{v\in V(G)}V(H_v).
\]
Hence \(\mathcal{V}(G\odot H)\) is **monic**, because there is exactly one mutual-visibility set of maximum size [2509.02509]. The main formula is
\[
\mathcal{V}(G\odot H) = \mathcal{V}(G) +\big((1+x)^{mn}-1\big) +mx\big(\mathcal{V}_2(H)-1\big) +\sum_Q p_Q(x),
\]
where \(|V(G)|=m>1\), \(|V(H)|=n\), and the \(p_Q(x)\) terms are governed by \(c_Q\)-visible sets and inclusion–exclusion [2509.02509].

On the algorithmic side, the direct computation problem is expensive. A known mutual-visibility test on \((G,P)\) runs in
\[
O\bigl(|P|(|V|+|E|)\bigr),
\]
and exhaustive enumeration over all subsets yields total time
\[
O\bigl(|V|(|V|+|E|)2^{|V|}\bigr),
\]
summarized as essentially \(O(n^3 2^n)\) in the worst case [2507.01851]. This computational cost explains the emphasis on structural formulas for graph classes and graph products [2509.22571].

## 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" [1501.02357], visibility is formulated as a relation \(v\) 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 \(v\) has a local Leader uniform topology. That paper **does not define a visibility polynomial** [1501.02357].

Similarly, "On the approximability of graph visibility problems" [2407.00409] 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 [2407.00409].

A second source of ambiguity arises in lattice-point visibility. In "Visibility of Lattice Points across Polynomials" [2601.15877], one studies visibility along the polynomial family
\[
\mathcal{F}(a_n,\dots,a_1)=\left\{\,y=q(a_nx^n+\cdots+a_1x)\mid q\in\mathbb{Q}^+\right\},
\]
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" [2604.23050], 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 [2601.15877], [2604.23050].

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.

Source: https://www.emergentmind.com/topics/visibility-polynomial