For a given simple graph $G$ with $n$ vertices $v\_1,v\_2,\dots v\_n$, the corresponding degree sequence is $d\_1,d\_2,\cdots,d\_n$. My qusetion is:



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> How to determine whether there exist subgraphs in $G$ such that it has the degree sequence of $d\_1',d\_2'\cdots,d\_n'$ (where $d\_i'\le d\_i$)? If exist, how to get it?
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(This is an extension of the question at [here](https://math.stackexchange.com/questions/991607/existence-of-graphs-when-given-the-degrees-of-all-vertices).)



Existence of subgraphs when given its degree sequence
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.


