Figure 11: A walk going back and forth along a path of $R$, which gives rise to $U$-turns.
These two lemmas complete the proof: we can indeed generate a collection of $2^{\mathcal{O}(k^2)}$ pushed weak linkages containing all canonical pushed weak linkages using only $2^{\mathcal{O}(k)}$ copies of any edge.
3 Preliminaries
Let $A$ be a set of elements. A cyclic ordering $\prec$ on $A$ is an ordering $(a_0, a_1, \ldots, a_{|A|-1})$ of the elements in $A$ such that, by enumerating $A$ in clockwise order starting at $a_i \in A$, we refer to the ordering $a_i, a_{(i+1)} \bmod |A|, \ldots, a_{(i+|A|-1)} \bmod |A|$, and by enumerating $A$ in counter-clockwise order starting at $a_i \in A$, we refer to the ordering $a_i, a_{(i-1)} \bmod |A|, \ldots, a_{(i-|A|+1)} \bmod |A|$. We consider all cyclic orderings of $A$ that satisfy the following condition to be the equivalent (up to cyclic shifts): the enumeration of $A$ in cyclic clockwise order starting at $a_i$, for any $a_i \in A$, produces the same sequence. For a function $f : X \to Y$ and a subset $X' \subseteq X$, we denote the restriction of $f$ to $X'$ by $f|_{X'}$.
Graphs. Given an undirected graph $G$, we let $V(G)$ and $E(G)$ denote the vertex set and edge set of $G$, respectively. Similarly, given a directed graph (digraph) $D$, we let $V(D)$ and $A(D)$ denote the vertex set and arc set of $D$, respectively. Throughout the paper, we deal with graphs without self-loops but with parallel edges. Whenever it is not explicitly written otherwise, we deal with undirected graphs. Moreover, whenever $G$ is clear from context, denote $n = |V(G)|$.
For a graph $G$ and a subset of vertices $U \subseteq V(G)$, the subgraph of $G$ induced by $U$, denoted by $G[U]$, is the graph on vertex set $U$ and edge set ${ {u, v} \in E(G) : u, v \in U }$. Additionally, $G-U$ denotes the graph $G[V(G) \setminus U]$. For a subset of edges $F \subseteq E(G)$, $G-F$ denotes the graph on vertex set $V(G)$ and edge set $E(G) \setminus F$. For a vertex $v \in V(G)$, the set of neighbors of $v$ in $G$ is denoted by $N_G(v)$, and for a subset of vertices $U \subseteq V(G)$, the open neighborhood of $U$ in $G$ is defined as $N_G(U) = \bigcup_{v \in U} N_G(v) \setminus U$. Given three subsets of vertices $A, B, S \subseteq V(G)$, we say that S separates $A$ from $B$ if $G-S$ has no path with an endpoint in $A$ and an endpoint in $B$. For two vertices $u, v \in V(G)$, the distance between $u$ and $v$ in $G$ is the length (number of edges) of the shortest path between $u$ and $v$ in $G$ (if no such path exists, then the distance is $\infty$), and it is denoted by $\text{dist}_G(u, v)$; in case $u = v$, $\text{dist}_G(u, v) = 0$. For two subsets $A, B \subseteq V(G)$, define $\text{dist}G(A, B) = \min{u \in A, v \in B} \text{dist}_G(u, v)$. A linkage of order $k$ in $G$ is an ordered family $\mathcal{P}$ of $k$ vertex-disjoint paths in $G$. Two linkages $\mathcal{P} = (P_1, \ldots, P_k)$ and $\mathcal{Q} = (Q_1, \ldots, Q_k)$ are aligned if for all $i \in {1, \ldots, k}$, $P_i$ and $Q_i$ have the same endpoints.
For a tree $T$ and $d \in \mathbb{N}$, let $V_{\ge d}(T)$ (resp. $V_d(T)$) denote the set of vertices of degree at least (resp. exactly $d$ in $T$. For two vertices $u, v \in V(T)$, the unique subpath of $T$ between $u$ and $v$ is denoted by $\textbf{path}T(u, v)$. We say that two vertices $u, v \in V(T)$ are near each other if $\textbf{path}T(u, v)$ has no internal vertex from $V{\ge 3}(T)$, and call $\textbf{path}T(u, v)$ a degree-2 path. In case $u, v \in V{\ge 3}(T) \cup V{=1}(T)$, $\textbf{path}_T(u, v)$ is called a maximal degree-2 path.
Planarity. A planar graph is a graph that can be embedded in the Euclidean plane, that is, there exists a mapping from every vertex to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped to the endpoints of the corresponding edge, and all curves are disjoint except on their extreme points. A plane graph $G$ is a planar graph with a fixed embedding. Its faces are the regions bounded by the edges, including the outer infinitely large region. For every vertex $v \in V(G)$, we let $E_G(v) = (e_0, e_1, \ldots, e_{t-1})$ for $t \in \mathbb{N}$ where $e_0, e_1, \ldots, e_{t-1}$ are the edges incident to $v$ in clockwise order (the decision which edge is $e_0$ is arbitrary). A planar graph $G$ is triangulated if