Let $G$ have vertices $v_1,\dots,v_n$. Let $Z(G)$ denote the convex
hull in $\mathbb{R}^n$ of all *ordered* degree sequences
$(\deg(v_1),\dots ,\deg(v_n))$ of subgraphs of $G$. One way of stating
the well-known Erdős-Gallai characterization of degree sequences of
spanning subgraphs of the complete graph $K_n$ is that every integer
point in $Z(G)$ with even coordinate sum is a degree sequence of a
spanning subgraph of $K_n$. We can then ask for what other graphs $G$
is this statement true. A complete characterization is due to
Fulkerson, Hoffman, and McAndrew, Some properties of graphs with
multiple edges, *Canad. J. Math.* **17** (1965), 166-177, namely, no
induced subgraph of $G$ consists of two vertex-disjoint odd cycles
(with no other edges). Equivalently, every induced subgraph of $G$ has
at most one nonbipartite component.