How to prove that a minimum edge cover is a forest?
A forest is a graph in which every component is a tree. If there is only one component, and that component is a tree, the graph is a forest.

So why is every minimum edge cover a forest? Suppose you have a graph $G=(V,E)$ and an edge cover&nbsp;$C\subseteq E$. Suppose some component of the graph $(V,C)$ contains a cycle, say $x_1x_2\dots x_kx_1$. Then $C\setminus\{x_1x_2\}$ is a smaller edge cover than&nbsp;$C$: $x_1$&nbsp;is covered by the edge&nbsp;$x_kx_1$, $x_2$&nbsp;is covered by&nbsp;$x_2x_3$ and every other vertex in&nbsp;$V$ is still covered by whatever covered it in&nbsp;$C$. Therefore, every minimum edge cover must contain no cycles, and an undirected graph that contains no cycles is a forest.

Indeed, we can go even farther. If an edge cover&nbsp;$C$ contains a path $x_1x_2x_3x_4$ then $C\setminus\{x_2x_3\}$ is a smaller edge cover, by essentially the same reasoning as above. Therefore any minimal edge cover is a forest that contains no path of three or more edges. So, not only is every minimum edge cover a forest, but it's a forest where every component is a star (a complete bipartite graph $K_{1,k}$ for some $k\geq 1$; note that $K_{1,1}$ is an edge and $K_{1,2}$ is a two-edge path).