Visibility Polynomial

Let be a simple graph. If denotes the number of mutual-visibility sets of of size , and is the mutual-visibility number, the visibility polynomial is defined by

(1)

(Bujtása et al. 2024, Tonny and Shikhi 2025). Here, denotes the mutual-visibility number, not the used for the matching polynomial, circuit rank, or a strongly regular graph parameter.

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)
		${(1+x)^2 for n=1; (1+x)^(2n)-2[x(1+x)]^n+x^n(2+2nx+x^n) otherwise$	(6)
			(7)
			(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 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).

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.

Tonny, K. B. and Shikhi, M. "On the Visibility Polynomial of Graphs." 2 Jul 2025. https://arxiv.org/abs/2507.01851.

Visibility Polynomial

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