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Choose a contractionApply the homotopy extension property to the initial map ${\mathrm{id}}_X:X\to X$ and to $h$, regarded as a homotopy into $X$. It suppliessuch that

At $t=1$, the map $H_1$ sends every point of $A$ to $a_0$. It is therefore constant on the equivalence classes of the quotient topology $X/A$. By the universal property of the quotient topology, there is a continuous mapsuch thatwhere $q:X\to X/A$ is the quotient map. The homotopy $H$ immediately gives

For every $t$, the map $q\circ H_t$ is constant on $A$, because $H(a,t)=h(a,t)\in A$. It therefore descends toThis descended map is continuous: the product $q\times {\mathrm{id}}_I$ is a quotient map because $I$ is compact Hausdorff, and $q\circ H$ is constant on its fibres.
At the endpoints,ThusThe maps $q$ and $g$ are homotopy inverses, soThis proves the theorem on collapsing a contractible cofibration.

Choose a spanning tree $T$ of the finite connected graph $X$. It contains every vertex of $X$ and has $v(X)-1$ edges by part (i). Collapsing $T$ to a point is a homotopy equivalence: $T$ is a contractible subcomplex, and collapsing a contractible cofibration in a CW complex preserves homotopy type.

Every edge outside $T$ then becomes a circle attached to the common point. Hence the quotient space $X/T$ is a bouquet of circles withpetals. The fundamental group of a bouquet of circles is the free group with one member of a free basis of a group for each petal. Therefore the rank of the fundamental group of a finite connected graph isMore concretely, choose a vertex as base point; follow the unique path in a graph in $T$ to one end of a non-tree edge, traverse it, and return through $T$. These based loops give the free basis of a group. Thus $r$ is the number of free group generators, equivalently the minimum size of a generating set of a group, rather than the size of every possible redundant generating set of a group. Minimality also follows by abelianization, since $F_r$ has abelianization ${\mathbb{Z}}^r$.