  We define a weakly threshold sequence to be a degree sequence
$d=(d_1,\dots,d_n)$ of a graph having the property that $\sum_{i \leq k} d_i
\geq k(k-1)+\sum_{i > k} \min\{k,d_i\} - 1$ for all positive $k \leq
\max\{i:d_i \geq i-1\}$. The weakly threshold graphs are the realizations of
the weakly threshold sequences. The weakly threshold graphs properly include
the threshold graphs and satisfy pleasing extensions of many properties of
threshold graphs. We demonstrate a majorization property of weakly threshold
sequences and an iterative construction algorithm for weakly threshold graphs,
as well as a forbidden induced subgraph characterization. We conclude by
exactly enumerating weakly threshold sequences and graphs.
