Let 
be a simple graph on vertex set 
and and let 
be a subset 
. Call two vertices 
"
-visible" if there exists a shortest path from 
to 
such that none of the internal vertices on the path belong to the set 
. A set 
is then called a mutual-visibility set of 
if every pair of vertices in 
is 
-visible (Tonny and Shikhi 2025).
The cardinality of the largest mutual-visibility set of 
is called the mutual-visibility number of 
, denoted 
(not to be confused with the matching-generating polynomial 
), and the number of such mutual-visibility sets of 
is denoted 
(Tonny and Shikhi 2025). Letting 
denote the number of mutual visibility sets of size 
, the visibility polynomial 
is then defined by
 | (1) |
(Bujtása et al. 2024, Tonny and Shikhi 2025).
Since every set of at most two vertices in a connected graph on 
vertices forms a mutual-visibility set, the visibility polynomial of a connected graph 
always begins
 | (2) |
(Tonny and Shikhi 2025), where 
is a binomial coefficient.
Special cases include
 |  |  | (3) |
 |  |  | (4) |
 |  |  | (5) |
 |  | ![{(1+x)^2 for n=1; (1+x)^(2n)-2[x(1+x)]^n+x^n(2+2nx+x^n) otherwise](https://mathworld.wolfram.com/images/equations/VisibilityPolynomial/Inline37.svg) | (6) |
 |  |  | (7) |
 |  |  | (8) |
where
 | (9) |
A formula for 
with 
involving three sums is given by Tonny and Shikhi (2025).
The visibility polynomial of a disconnected graph with 
components 
, ..., 
is given by
 | (10) |
(Tonny and Shikhi 2025).
