A subgraph is called spanning when it includes all of the vertices of the given graph.
Is there a term for a subgraph which includes all the edges of a graph?
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If the subgraph includes all edges of the original graph, then it also includes all its vertices, except maybe some with degree $0$.
The minimal such subgraph is the graph in which all $0$-degree vertices were removed. It is called the $1$-core of the graph, a very special case of $k$-core.