to [@west] for any undefined definitions and notation. A graph $G$ has vertex set $V(G)$, edge set $E(G)$, and maximum degree $\Delta(G)$. If a vertex $v$ has degree $j$ we refer to it as a [*$j$-vertex*]{}, and if $v$ has a neighbor that is a $j$-vertex, we say it is a [*$j$-neighbor*]{} of $v$. When $G$ is planar we let $F(G)$ denote the set of faces of $G$, and $\ell(f)$ denote the length of a face $f$. The [*girth*]{} of a graph $G$ is length of its shortest cycle. A graph $G$ is $\{a,b\}$-regular if for every $v$ in $G$, the degree of $v$ is either $a$ or $b$. Every graph $G$ with maximum degree $d$ is contained in a prescribed $\{1,d\}$-regular graph, denoted $\operatorname{ex}_d(G)$, the [*$d$-expansion*]{} of $G$. To construct $\operatorname{ex}_d(G)$, add $d-d(v)$ pendant edges to each vertex $v$ in $G$ where $d(v) \in \{2,\dots, d\}$. Additionally, let the [*contracted graph*]{} of $G$, denoted $\operatorname{ct}(G)$ be the graph obtained by deleting all 1-vertices of $G$. A vertex $v$ in $G$ is a *$2{{\ensuremath{^\perp}}}$-vertex* if $v$ is a 2-vertex in $\operatorname{ct}(G)$. Thus, for the remainder of the paper a vertex $v$ is a *$k^+$-vertex* in $G$ if it has degree 