If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is an edge cover if $C\cap e \neq \emptyset$ for all $e\in E$.
The "best" covers in some sense are subsets $C\subseteq V$ that meet every edge in exactly one point - but in many graphs, such a nice cover does not exist; there are often "bad" edges $e$ so that $C$ covers both points of the edge (i.e. $e\subseteq C$). We define the set of bad edges with respect to an edge cover $C$ by $$\text{Bad}(C) = \{e\in E: e\subseteq C\};$$ the other edges are good with respect to $C$, that is we set $\text{Good}(C) = E\setminus\text{Bad}(C)$.
Question. Is there a connected, infinite graph $G=(V,E)$ such that for all edge covers $C$ we have $|\text{Good}(C)| < |E|$?