  A graphic sequence $\pi$ is potentially $H$-graphic if there is some
realization of $\pi$ that contains $H$ as a subgraph. The
Erd\H{o}s-Jacobson-Lehel problem asks to determine $\sigma(H,n)$, the minimum
even integer such that any $n$-term graphic sequence $\pi$ with sum at least
$\sigma(H,n)$ is potentially $H$-graphic. The parameter $\sigma(H,n)$ is known
as the potential function of $H$, and can be viewed as a degree sequence
variant of the classical extremal function ${\rm ex}(n,H)$. Recently, Ferrara,
LeSaulnier, Moffatt and Wenger [On the sum necessary to ensure that a degree
sequence is potentially $H$-graphic, Combinatorica 36 (2016), 687--702]
determined $\sigma(H,n)$ asymptotically for all $H$, which is analogous to the
Erd\H{o}s-Stone-Simonovits Theorem that determines ${\rm ex}(n,H)$
asymptotically for nonbipartite $H$.
  In this paper, we investigate a stability concept for the potential number,
inspired by Simonovits' classical result on the stability of the extremal
function. We first define a notion of stability for the potential number that
is a natural analogue to the stability given by Simonovits. However, under this
definition, many families of graphs are not $\sigma$-stable, establishing a
stark contrast between the extremal and potential functions. We then give a
sufficient condition for a graph $H$ to be stable with respect to the potential
function, and characterize the stability of those graphs $H$ that contain an
induced subgraph of order $\alpha(H)+1$ with exactly one edge.
