Stallings Theorem About Ends of Groups - Ends of Graphs

Ends of Graphs

Let Γ be a connected graph where the degree of every vertex is finite. One can view Γ as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of Γ are the ends of this topological space. A more explicit definition of the number of ends of a graph is presented below for completeness.

Let n ≥ 0 be a non-negative integer. The graph Γ is said to satisfy e(Γ) ≤ n if for every finite collection F of edges of Γ the graph Γ − F has at most n infinite connected components. By definition, e(Γ) = m if e(Γ) ≤ m and if for every 0 ≤ n < m the statement e(Γ) ≤ n is false. Thus e(Γ) = m if m is the smallest nonnegative integer n such that e(Γ) ≤ n. If there does not exist an integer n ≥ 0 such that e(Γ) ≤ n, put e(Γ) = ∞. The number e(Γ) is called the number of ends of Γ.

Informally, e(Γ) is the number of "connected components at infinity" of Γ. If e(Γ) = m < ∞, then for any finite set F of edges of Γ there exists a finite set K of edges of Γ with F ⊆ K such that Γ − F has exactly m infinite connected components. If e(Γ) = ∞, then for any finite set F of edges of Γ and for any integer n ≥ 0 there exists a finite set K of edges of Γ with F ⊆ K such that Γ − K has at least n infinite connected components.