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Visibility Polynomial


Let G be a simple graph on vertex set V and and let X be a subset V. Call two vertices u,v in V "X-visible" if there exists a shortest path from u to v such that none of the internal vertices on the path belong to the set X. A set X is then called a mutual-visibility set of G if every pair of vertices in X is X-visible (Tonny and Shikhi 2025).

The cardinality of the largest mutual-visibility set of G is called the mutual-visibility number of G, denoted mu(G) (not to be confused with the matching-generating polynomial mu(x)), and the number of such mutual-visibility sets of G is denoted r_mu(G) (Tonny and Shikhi 2025). Letting r_k denote the number of mutual visibility sets of size k, the visibility polynomial V(G) is then defined by

 V(G)=sum_(k=0)^mur_kx^k
(1)

(Bujtása et al. 2024, Tonny and Shikhi 2025)

Since every set of at most two vertices in a connected graph on n vertices forms a mutual-visibility set, the visibility polynomial of a connected graph G always begins

 V(G)=1+nx+(n; 2)x^2+...
(2)

(Tonny and Shikhi 2025), where (n; k) is a binomial coefficient.

Special cases include

V(K^__n)=1+nx
(3)
V(C_n)=1+nx+(n; 2)x^2+r_3(C_n)x^3
(4)
V(K_n)=(1+x)^n
(5)
V(K_(n,n))={(1+x)^2 for n=1; (1+x)^(2n)-2[x(1+x)]^n+x^n(2+2nx+x^n) otherwise
(6)
V(K_(1,n))=x+nx^2(1+x)^n
(7)
V(P_n)=1+nx+(n; 2)x^2
(8)

where

 r_3(C_3)={(n(n^2-1))/(24)   for n odd; ((n-2)n(n+8))/(24)   for n even.
(9)

A formula for V(K_(m,n)) with 3<=m<=n involving three sums is given by Tonny and Shikhi (2025).

The visibility polynomial of a disconnected graph with c components G_1, ..., G_c is given by

 V(G)=V(G_1)+...+V(G_c)-c+1
(10)

(Tonny and Shikhi 2025).


See also

Shortest Path

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References

Bujtása, C.; Klavžara, S.; and Tiand, J. "Visibility Polynomials, Dual Visibility Spectrum, and Characterization of Total Mutual-Visibility Sets." 4 Dec 2024. https://arxiv.org/abs/2412.03066.Di Stefano, G. "Mutual Visibility in Graphs." Appl. Math. Comput. 419, 126850, 2022.Tonny, K. B. and Shikhi, M. "On the Visibility Polynomial of Graphs." 2 Jul 2025. https://arxiv.org/abs/2507.01851.

Cite this as:

Weisstein, Eric W. "Visibility Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VisibilityPolynomial.html