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 $|\text{Good}(C)| < |E|$?