  A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which
parallel edges and loops are permitted) with $V=\{1,\ldots,n\}$ and
$\Sigma\subseteq E$. The edges in $\Sigma$ are called odd edges and the other
edges of $E$ even. By $S(G,\Sigma)$ we denote the set of all symmetric $n\times
n$ real matrices $A=[a_{i,j}]$ such that if $a_{i,j} < 0$, then there must be
an even edge connecting $i$ and $j$; if $a_{i,j} > 0$, then there must be an
odd edge connecting $i$ and $j$; and if $a_{i,j} = 0$, then either there must
be an odd edge and an even edge connecting $i$ and $j$, or there are no edges
connecting $i$ and $j$. (Here we allow $i=j$.) For a symmetric real matrix $A$,
the partial inertia of $A$ is the pair $(p,q)$, where $p$ and $q$ are the
number of positive and negative eigenvalues of $A$, respectively. If
$(G,\Sigma)$ is a signed graph, we define the \emph{inertia set} of
$(G,\Sigma)$ as the set of the partial inertias of all matrices $A \in
S(G,\Sigma)$.
  In this paper, we present a formula that allows us to obtain the minimal
elements of the inertia set of $(G,\Sigma)$ in case $(G,\Sigma)$ has a
$1$-separation using the inertia sets of certain signed graphs associated to
the $1$-separation.
