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combinatorics / Graph polynomials

The Integer Domination Root Conjecture

Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial $D(G, x)$, proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at $x = -4$, built from an S-unit branch cancellation mechanism.

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combinatoricsJul 31, 2026Significance 11/100Registry: unreviewed

The Integer Domination Root Conjecture

Prior state unknown→disproved

Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial $D(G, x)$, proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at $x = -4$, built from an S-unit branch cancellation mechanism.

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Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial $D(G, x)$, proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at $x = -4$, built from an S-unit branch cancellation mechanism.

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