strong $k$-edge-coloring of $G-D'$ can be extended to a strong $k$-edge-coloring of $G$. One necessary property for the selection of $D'$ is that no two edges that remain in $G-D'$ can have distance at most two in $G$ but distance strictly larger than two in $G - D'$. In the next subsection we describe several reducible configurations.

Reducible Configurations
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This subsection contains description of four types of reducible configurations. Each configuration is described in terms of how it appears within $\operatorname{ct}(G)$ where $G$ is a graph with maximum degree $\Delta(G) \leq d$ for some $d \geq 4$.

Let $t$ be a positive integer. The *$t$-caterpillar* is formed by two $3^+$-vertices $v_0$ and $v_{t+1}$ with a path $v_0v_1\dots v_tv_{t+1}$ where each $v_i$ is a $2{{\ensuremath{^\perp}}}$-vertex for every $i \in \{1,\dots,t\}$.

Let $t_1,\dots, t_k$ be nonnegative integers. A configuration $Y(t_1,\dots,t_k)$ is formed by a $k^+$-vertex $v$ and $k$ internally disjoint paths of lengths $t_1+1,\dots,t_k+1$ with $v$ as a common endpoint, where the internal vertices of the paths are $2{{\ensuremath{^\perp}}}$-vertices. We call such configuration a *$Y$-type configuration about $v$*, see Figure \[fig:Y\].

A configuration $H(t_1,t_2;r;s_1,s_2)$ is formed by two 3-vertices $u$ and $v$ and 5 internally disjoint paths of lengths $t_1+1$, $t_2+1$, 