  For a positive integer $n$, the set of all integers greater than or equal to
$n$ is denoted by $\mathcal T(n)$. A sum of generalized $m$-gonal numbers $g$
is called tight $\mathcal T(n)$-universal if the set of all nonzero integers
represented by $g$ is equal to $\mathcal T(n)$. In this article, we prove the
existence of a minimal tight $\mathcal T(n)$-universality criterion set for a
sum of generalized $m$-gonal numbers for any pair $(m,n)$. To achieve this, we
introduce an algorithm giving all candidates for tight $\mathcal
T(n)$-universal sums of generalized $m$-gonal numbers for any given pair
$(m,n)$. Furthermore, we provide some experimental results on the
classification of tight $\mathcal T(n)$-universal sums of generalized $m$-gonal
numbers.
