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arXiv · 2609.06569

Generalizations and Applications of the Brown--Tufts Lemma

Pingping Shao·Chengye Zhao
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Abstract

Brown and Tufts proved in 2014 the substitution formula for the domination polynomial under graph substitution (the lexicographic product), and used it to show that the closure of the domination roots is the whole complex plane. In this paper we undertake a systematic study of the extent to which this lemma extends to the most common domination variants. We first prove that the connected domination polynomial satisfies exactly the same substitution formula. We then show that the formula fails for both independent domination and total domination: for independent domination we give a counterexample and establish a corrected formula, and for total domination we exhibit a concrete counterexample. Finally, applying the substitution formula for the connected domination polynomial, we prove that the closure of the real connected domination roots is $(-\infty,0]$, and that the closure of the connected domination roots is the whole complex plane.

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Pingping Shao, Chengye Zhao. 2026-09-06. Generalizations and Applications of the Brown--Tufts Lemma. https://arxiv.org/abs/2609.06569

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