 follow the usual conventions of nonstandard analysis. ${I \kern -4.5pt N}= \{0,1,2,\ldots\}$ is the set of natural numbers, and $^{*}\! {I \kern -4.5pt N}$ is the set of hypernaturals. The standard hypernaturals are (i.e., can be identified with) the natural numbers. Also, $\langle a_{n}\rangle$ or $\langle a_{n}\!: n\in{I \kern -4.5pt N}\rangle$ or $\langle a_{0},a_{1},
a_{2},\ldots\rangle$ denotes a sequence whose elements can be members of any set, such as the set $X$ of nodes in a conventional graph $G=\{X,B\}$, where $B$ is the set of branches, a branch being a two-element set of nodes. On the other hand, $[a_{n}]$ denotes an equivalence class of sequences, where two sequences $\langle a_{n}\rangle$ and $\langle b_{n}\rangle$ are taken to be equivalent if $\{n\!:a_{n}=b_{n}\}\in {\cal F}$, where ${\cal F}$ is any chosen and fixed free ultrafilter.[^1] The $a_{n}$ appearing in $[a_{n}]$ are understood to be the elements of any one of the sequences in the equivalence class. At times, we will use the more specific notation $[\langle a_{0},a_{1},a_{2},\ldots\rangle]$. More generally, we adhere to the notations and terminology appearing in [@go].

The ordinals are denoted in the usual way: $\omega$ is the first transfinite ordinal. With $\tau\in{I \kern -4.5pt N}$, the product $\omega\cdot \tau$ is the sum of $\tau$ summands, each being $\omega$.

The Nonstandard Enlargement of a Graph
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Throughout Sections 2 to 4, we assume that the conventionally infinite graph $G$ is connected and has infinitely many nodes. The definition of a nonstandard graph that we use herein is given in [@gn Section 8.1], a special case of which is the “enlargement” of a graph $G$.

Let us define the [*enlargement*]{} $^{*}\!G$ of $G$ here as well in order to remove any need for referring to [@gn]. $G=\{X,B\}$ is now taken to be a conventional connected graph having an infinite set $X$ of nodes and therefore an infinite set of branches as well, each branch being a two-element set of nodes. Thus, 