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)
| |||
(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).