  We present an algorithm to test whether a given graphical degree sequence is
forcibly connected or not and prove its correctness. We also outline the
extensions of the algorithm to test whether a given graphical degree sequence
is forcibly $k$-connected or not for every fixed $k\ge 2$. We show through
experimental evaluations that the algorithm is efficient on average, though its
worst case run time is probably exponential. We also adapt Ruskey et al's
classic algorithm to enumerate zero-free graphical degree sequences of length
$n$ and Barnes and Savage's classic algorithm to enumerate graphical partitions
of even integer $n$ by incorporating our testing algorithm into theirs and then
obtain some enumerative results about forcibly connected graphical degree
sequences of given length $n$ and forcibly connected graphical partitions of
given even integer $n$. Based on these enumerative results we make some
conjectures such as: when $n$ is large, (1) almost all zero-free graphical
degree sequences of length $n$ are forcibly connected; (2) almost none of the
graphical partitions of even $n$ are forcibly connected.
