 Edge support of a graph

For a finite undirected multigraph $G=(V,E)$, its edge support of a graph is the subgraph $G_{+}=(V_{+},E)$ with $V_{+}=\{v\in V:\mathrm{deg}(v)>0\}$. It removes exactly the isolated vertices, retaining every edge, including self-loops and parallel edges. When $E\ne \varnothing$, connectivity of this subgraph is the relevant condition for an Eulerian trail or Euler circuit.