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Source interface. Section 1, printed p. 147, and the discussion of graph decompositions in Sections 2 and 4 (published PDF). This expands the elementary graphic example used by the source, not a separate numbered theorem or a reconstruction of its cited graph-theoretic papers.

Statement. Let $G$ be a finite loopless graph with fixed vertex set $V$ and edge set $E$. The acyclic edge sets form a matroid on $E$. If $c(A)$ is the number of connected components of $(V,A)$, counting isolated vertices, its rank is

$$r(A)=|V|-c(A).$$

Consequently, for $k\ge1$:

1. $E$ partitions into $k$ forests, with empty parts allowed, exactly when $|A|\le k(|V|-c(A))$ for all $A\subseteq E$.
2. There are $k$ edge-disjoint spanning forests, each spanning every connected component of $G$, exactly when

$$|E\setminus A|\ge k(c(A)-c(E)) \qquad(A\subseteq E).$$

For connected $G$, these bases are spanning trees.

Proof. A subset of a forest is a forest. If $F\subseteq A$ is a maximal forest, its connected components are exactly those of $(V,A)$. Otherwise a path in $(V,A)$ between two different components of $(V,F)$ supplies an edge joining different forest components, which could be added without creating a cycle.

A finite forest with vertex set $V$ and $c$ components has $|V|-c$ edges. Each nontrivial finite tree has a leaf: an endpoint of a longest path cannot have an additional neighbor without either extending the path or creating a cycle. Thus one can remove a leaf and its incident edge repeatedly from every nontrivial tree, leaving one vertex per component; each removal lowers both the edge and vertex counts by one. The formula also holds for the empty vertex set. Thus every maximal forest in $A$ has $|V|-c(A)$ edges, proving the matroid axiom and rank.

Apply Theorems 1 and 2 with this rank. The first criterion is immediate. In the second, $r(E)-r(A)=c(A)-c(E)$, giving the displayed inequality. A base is exactly a forest with the same components as $G$, as the maximality argument showed. $\square$

No numbered Erdős-problem implication or current graph bound is inferred here. The exact equivalence to other vertex-partition formulations in the cited Tutte and Edmonds papers remains outside this source compilation.