Sometimes, many links in a multiple CPU system may be missing. A missing edge implies that a link between 2 CPUs that was faulty. The existence of missing edges in a system may affect the diagnosability of the whole system, and degrees and the local diagnosability of some nodes. Especially, in a regular graph, nodes that are adjacent to missing edges have lower degrees than others. Hence, those nodes may not keep the strong local diagnosability feature, and the graph may not keep the strong local diagnosability feature again. Those new degrees can be used to determine whether the incomplete graph keeps the strong local diagnosability feature or not. Next, we demonstrate that an n-dimensional wheel network $C{W}_{n}(n\ge 6)$ keeps the strong local diagnosability feature even with up to $2n-4$ missing edges.

**Proof.** By Proposition 4, $C{W}_{n}$ is node transitive. Therefore, $\left(1\right)$ can be chosen as the root of an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$, i.e., $x=\left(1\right)$, where $\left(1\right)$ is the identity element of the symmetric group ${S}_{n}$. $C{W}_{n}$ can be partitioned into n disjoint subgraphs $C{W}_{n}^{1},\phantom{\rule{4pt}{0ex}}C{W}_{n}^{2},\cdots ,C{W}_{n}^{n}$, where each node $u={u}_{1}{u}_{2}\cdots {u}_{n}\in V\left(C{W}_{n}^{i}\right)$ has a fixed integer i in the last position ${u}_{n}$ for $i\in \left[n\right]$. Clearly, $C{W}_{n}^{i}$ is isomorphic to $B{S}_{n-1}$, where $B{S}_{n-1}$ is the bubble-sort star graph with the dimension $n-1$.

Let ${F}_{e}^{i}={F}_{e}\cap E\left(C{W}_{n}^{i}\right)$ for $1\le i\le n$, and ${F}^{*}={F}_{e}\cap (E\left(C{W}_{n}\right)\backslash {\sum}_{i=1}^{n}E\left(C{W}_{n}^{i}\right))$, then ${F}_{e}={F}^{*}\cup {F}_{e}^{1}\cup \cdots \cup {F}_{e}^{n}$. For convenience, let $\tilde{F}={F}^{*}\cap \{xu,xv,xw\}$, and denote $u=\left(1n\right)$, $v=\left(2n\right)$ and $w=(n-1,n)$ three outside neighbors of $\left(1\right)$. To prove this lemma, we need to discuss whether $\tilde{F}$ is an empty set or not. When $\tilde{F}=\varnothing $, the first step is to discover an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{n}}\left(x\right))$ in $C{W}_{n}^{n}-{F}_{e}^{n}$; the second step is to find a 3-path ${P}_{u}$ in $C{W}_{n}^{1}$, a 3-path ${P}_{v}$ in $C{W}_{n}^{2}$ and a 3-path ${P}_{w}$ in $C{W}_{n}^{n-1}$; the third step is to connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Then, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). For the case of $\tilde{F}\ne \varnothing $, we obtain an extended star to satisfy the lemma by removing any 4-path that contains any link of $\tilde{F}$ and starts at x from B.

It should be noted that removing a 4-path P from a graph G means removing all nodes and links of the 4-path except y from a graph G in this paper, where y is the only common node of P and G.

Firstly, we give two claims as follows.

**Claim** **1.** If $|{F}_{e}^{i}|\le 2n-7$, then there exists an extended star $ES(z;{d}_{C{W}_{n}^{i}-{F}_{e}^{i}}\left(z\right))$ in $C{W}_{n}^{i}-{F}_{e}^{i}$ at $z$ for $z\in V\left(C{W}_{n}^{i}\right)$. In the extended star $ES(z;{d}_{C{W}_{n}^{i}-{F}_{e}^{i}}\left(z\right))$, there exist at least $(2n-5-|{F}_{e}^{i}\left|\right)$ and at most $(2n-5)$ 4-paths (resp. 3-paths) with just common node $z$.

**Proof.** Notice that each $C{W}_{n}^{i}$ is isomorphic to $B{S}_{n-1}$ and $|{F}_{e}^{i}|\le 2n-7=2(n-1)-5$, by Lemma 9, then there exists an extended star $ES(z;{d}_{C{W}_{n}^{i}-{F}_{e}^{i}}\left(z\right))$ in $C{W}_{n}^{i}-{F}_{e}^{i}$ at $z$ for $z\in V\left(C{W}_{n}^{i}\right)$ and ${d}_{C{W}_{n}^{i}}\left(z\right)=2(n-1)-3=2n-5$. Therefore, ${d}_{C{W}_{n}^{i}-{F}_{e}^{i}}\left(z\right)\ge 2n-5-\left|{F}_{e}^{i}\right|$, and we can find at least $(2n-5-|{F}_{e}^{i}\left|\right)$ and at most $(2n-5)$ 4-paths (resp. 3-paths) with just common node $z$ in the $ES(z;{d}_{C{W}_{n}^{i}-{F}_{e}^{i}}\left(z\right))$ combining the definition of the extended star. In particular, in the extended star $ES(z;{d}_{C{W}_{n}^{i}-{F}_{e}^{i}}\left(z\right))$, there exist just $(2n-5-|{F}_{e}^{i}\left|\right)$ 4-paths (resp. 3-paths) with just common node $z$ if and only if each of edges in ${F}_{e}^{i}$ is incident with $z$. □

**Claim** **2.** If $|{F}_{e}^{i}|<2n-5$, then there exists at least a 3-path in $C{W}_{n}^{i}-{F}_{e}^{i}$ starting at $z$ for $z\in V\left(C{W}_{n}^{i}\right)$.

**Proof.** By Propositions 1–3, we have that $\lambda \left(B{S}_{n-1}\right)=2(n-1)-3=2n-5$. Since each $C{W}_{n}^{i}$ is isomorphic to $B{S}_{n-1}$, we have that $\lambda \left(C{W}_{n}^{i}\right)=2n-5$. If $|{F}_{e}^{i}|<2n-5$, then $C{W}_{n}^{i}-{F}_{e}^{i}$ is connected. By Proposition 6 and $|V\left(C{W}_{n}^{i}\right)|-|{F}_{e}^{i}|>(n-1)!-(2n-5)\ge 113$ for $n\ge 6$. So a 3-path starting at $z$ can be found in $ES(z;{d}_{C{W}_{n}^{i}-{F}_{e}^{i}}\left(z\right))$ for $z\in V\left(C{W}_{n}^{i}\right)$. □

Then we can find the extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ by discussing $|{F}_{e}|$ and $|{F}_{e}^{i}|$ as follows.

Case 1. $|{F}_{e}^{n}|\le 2n-7$.

By Claim 1, there exists an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{n}}\left(x\right))=A$ in $C{W}_{n}^{n}-{F}_{e}^{n}$ at x.

Case 1.1. $|{F}_{e}|<2n-5$.

Here, $|{F}_{e}^{i}|<2n-5$ for $i=1,2,n-1$. By Claim 2, there exists a 3-path ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) in $C{W}_{n}^{1}-{F}_{e}^{1}$ (resp. $C{W}_{n}^{2}-{F}_{e}^{2}$, $C{W}_{n}^{n-1}-{F}_{e}^{n-1}$). We connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any 4-path that contains any edge of $\tilde{F}$ and starts at x from B.

Case 1.2. $|{F}_{e}|=2n-5$.

If each of $|{F}_{e}^{1}|$, $|{F}_{e}^{2}|$ and $|{F}_{e}^{n-1}|$ is less than $2n-5$, then we can complete the proof as Case 1.1. Clearly, at most one of $|{F}_{e}^{1}|$, $|{F}_{e}^{2}|$ and $|{F}_{e}^{n-1}|$ is equal to $2n-5$, without loss of generality, we consider $|{F}_{e}^{1}|=2n-5$. Then $|{F}^{*}|+|{F}_{e}^{2}|+|{F}_{e}^{3}|+\cdots +|{F}_{e}^{n-1}|=0$. We choose a 4-path ${P}_{u}=\langle \left(1n\right),\left(1n\right)\left(2n\right),\left(1n\right)\left(2n\right)\left(23\right),\left(1n\right)\left(2n\right)\left(23\right)\left(34\right)\rangle $. Clearly, ${P}_{u}-u$ is in $C{W}_{n}^{2}$. By Claim 1 and $|{F}_{e}^{2}|=|{F}_{e}^{n-1}|=0$, there exist $(2n-5)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{2}$ (resp. $C{W}_{n}^{n-1}$). Notice that $v\notin V\left({P}_{u}\right)$, $|V\left({P}_{u}\right)\cap V\left(C{W}_{n}^{2}\right)|=3$ and $2n-5>3$ for $n\ge 6$, so there exists a 3-path ${P}_{v}$ starting at v in $C{W}_{n}^{2}$ such that ${P}_{u}$ and ${P}_{v}$ have no common node. We can choose a 3-path ${P}_{w}$ starting at w in $C{W}_{n}^{n-1}$. Notice that any two of ${P}_{u},{P}_{v}$ and ${P}_{w}$ have no common node, and each of ${P}_{u},{P}_{v}$ and ${P}_{w}$ does not contain a missing edge. We connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$.

Case 1.3. $|{F}_{e}|=2n-4$.

If each of $|{F}_{e}^{1}|$, $|{F}_{e}^{2}|$ and $|{F}_{e}^{n-1}|$ is less than $2n-5$, then we can complete the proof as Case 1.1. Clearly, at most one of $|{F}_{e}^{1}|$, $|{F}_{e}^{2}|$ and $|{F}_{e}^{n-1}|$ is equal to $2n-5$ or $2n-4$. Without loss of generality, let $|{F}_{e}^{1}|=2n-4$, or $2n-5$. If $|{F}_{e}^{1}|=2n-4$, then the proof for $|{F}_{e}^{1}|=2n-4$ can be completed as Case 1.2. If $|{F}_{e}^{1}|=2n-5$, then $|{F}^{*}|+|{F}_{e}^{2}|+|{F}_{e}^{3}|+\cdots +|{F}_{e}^{n-1}|=1$.

Case 1.3.1. $|{F}_{e}^{2}|+|{F}_{e}^{3}|+\cdots +|{F}_{e}^{n-1}|=1$.

Without loss of generality, let $|{F}_{e}^{2}|=1$. Here, $|{F}^{*}|+|{F}_{e}^{3}|+|{F}_{e}^{4}|+\cdots +|{F}_{e}^{n-1}|=0$. We choose a 4-path ${P}_{u}=\langle \left(1n\right),\left(1n\right)(n-1,n),\left(1n\right)(n-1,n)\left(23\right),\left(1n\right)(n-1,n)\left(23\right)\left(34\right)\rangle $. Clearly, ${P}_{u}-u$ is in $C{W}_{n}^{n-1}$. Combining Claim 1, there exist at least $(2n-6)$ 3-paths in $C{W}_{n}^{2}-{F}_{e}^{2}$ (resp. $C{W}_{n}^{n-1}-{F}_{e}^{n-1}$). Notice that $w\notin V\left({P}_{u}\right)$, $|V\left({P}_{u}\right)\cap V\left(C{W}_{n}^{n-1}\right)|=3$ and $2n-6>3$ for $n\ge 6$, so there exists a 3-path ${P}_{w}$ starting at w in $C{W}_{n}^{n-1}$ such that ${P}_{u}$ and ${P}_{w}$ have no common node. Choose a 3-path ${P}_{v}$ starting at v in $C{W}_{n}^{2}$. Notice that any two of ${P}_{u},{P}_{v}$ and ${P}_{w}$ have no common node, and each of ${P}_{u},{P}_{v}$ and ${P}_{w}$ does not contain a missing edge. We connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$.

Case 1.3.2. $|{F}_{e}^{2}|+|{F}_{e}^{3}|+\cdots +|{F}_{e}^{n-1}|=0$.

Notice that $|{F}^{*}|=1$. If $\tilde{F}=\varnothing $, then we choose a 4-path ${P}_{u}=\langle \left(1n\right),\left(1n\right)(n-1,n),\left(1n\right)(n-1,n)\left(23\right),\left(1n\right)(n-1,n)\left(23\right)\left(34\right)\rangle $. Clearly, ${P}_{u}-u$ is in $C{W}_{n}^{n-1}$. By Claim 1 and $|{F}_{e}^{2}|=|{F}_{e}^{n-1}|=0$, there exist at least $(2n-5)$ 3-paths in $C{W}_{n}^{2}-{F}_{e}^{2}$ (resp. $C{W}_{n}^{n-1}-{F}_{e}^{n-1}$). Notice that $w\notin V\left({P}_{u}\right)$, $|V\left({P}_{u}\right)\cap V\left(C{W}_{n}^{n-1}\right)|=3$ and $2n-5>3$ for $n\ge 6$, so there exists a 3-path ${P}_{w}$ starting at w in $C{W}_{n}^{n-1}$ such that ${P}_{u}$ and ${P}_{w}$ have no common node. Choose a 3-path ${P}_{v}$ starting at v in $C{W}_{n}^{2}$. Notice that any two of ${P}_{u},{P}_{v}$ and ${P}_{w}$ have no common node, and each of ${P}_{u},{P}_{v}$ and ${P}_{w}$ does not contain a missing edge. We connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B), and B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any 4-path that contains any link of $\tilde{F}$ and starts at x from B.

Case 2. $|{F}_{e}^{n}|=2n-6,|{F}_{e}|=2n-6$.

Here, $|{F}^{*}|+|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=0$. By Claim 1, we can choose a 3-path ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) from at least $(2n-5)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$, $C{W}_{n}^{n-1}$). Let f be an arbitrary link of ${F}_{e}^{n}$, and let ${F}_{e}^{{}^{\prime}}={F}_{e}^{n}\backslash \left\{f\right\}$. Then $|{F}_{e}^{{}^{\prime}}|=2n-7$, there exists an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$ in $C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}$ by Claim 1. Let ${A}^{{}^{\prime}}=ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$.

Case 2.1. $f\notin E\left({A}^{{}^{\prime}}\right)$.

Here, ${A}^{{}^{\prime}}$ is one of extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{n}}\left(x\right))$ at x of $C{W}_{n}^{n}-{F}_{e}^{n}$. Notice that $|{F}^{*}|=0$, so we can connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with ${A}^{{}^{\prime}}$, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ can be obtained in $C{W}_{n}-{F}_{e}$.

Case 2.2. $f\in E\left({A}^{{}^{\prime}}\right)$.

Let ${P}_{x}$ be a 4-path containing f and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ from ${A}^{{}^{\prime}}$, denoted by A.

Case 2.2.1. f is incident with x.

Notice that $|{F}^{*}|=0$, so we connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ can be obtained in $C{W}_{n}-{F}_{e}$.

Case 2.2.2. f is not incident with x.

Let ${P}_{a}$ be a 3-path containing f and starting at a, and then a is adjacent to x. Next, we consider $a=\left(1i\right)$ for $2\le i\le n-1$. Without loss of generality, let $a=\left(12\right)$. Notice that $|{F}_{e}^{2}|=0$, we can choose a 2-path ${P}_{{a}^{{}^{\prime}}}=\langle {a}^{{}^{\prime}},{a}^{{}^{\prime}}\left(23\right),{a}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ that does not contain a missing edge in $C{W}_{n}^{2}$, where ${a}^{{}^{\prime}}=\left(12\right)\left(1n\right)$. By Claim 1 and $|{F}_{e}^{i}|=0$, we can find $(2n-5)$ 3-paths in $C{W}_{n}^{i}$, where $i=1,2,n-1$. Note that$v\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$, $|V\left({P}_{{a}^{{}^{\prime}}}\right)\cap V\left(C{W}_{n}^{2}\right)|=3$ and $2n-5>3$ for $n\ge 6$, so we can find a 3-path ${P}_{v}$ in $C{W}_{n}^{2}$ that does not contain a missing edge, and ${P}_{v}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node. At the same time, we can find a 3-path ${P}_{u}$ (resp. ${P}_{w}$) in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{n-1}$), and connect ${P}_{{a}^{{}^{\prime}}}$ to a to obtain a 3-path ${P}_{a}$. Notice that each of ${P}_{a}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ do not contain a missing edge, and any two of them have no common node. We connect ${P}_{a}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ can be got in $C{W}_{n}-{F}_{e}$. The case of $a=(i,i+1)$ for $2\le i\le n-2$ can be proved similarly.

Case 3. $|{F}_{e}^{n}|=2n-6,|{F}_{e}|=2n-5$.

Notice that $|{F}^{*}|+|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=1$. By Claim 1, we can choose a 3-path ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) from at least $(2n-6)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$, $C{W}_{n}^{n-1}$). Let f be an arbitrary edge of ${F}_{e}^{n}$, and let ${F}_{e}^{{}^{\prime}}={F}_{e}^{n}\backslash \left\{f\right\}$. Then $|{F}_{e}^{{}^{\prime}}|=2n-7$. Combining Claim 1, there exists an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$ in $C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}$. Let ${A}^{{}^{\prime}}=ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$.

Case 3.1. $f\notin E\left({A}^{{}^{\prime}}\right)$.

Here, ${A}^{{}^{\prime}}$ is one of extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{n}}\left(x\right))$ at x in $C{W}_{n}^{n}-{F}_{e}^{n}$. We connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with ${A}^{{}^{\prime}}$, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|\le 1$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any edge of $\tilde{F}$ and starts at x from B.

Case 3.2. $f\in E\left({A}^{{}^{\prime}}\right)$.

Let ${P}_{x}$ be a 4-path containing f and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ from ${A}^{{}^{\prime}}$, denoted by A.

Case 3.2.1. f is incident with x.

Connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|\le 1$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any edge of $\tilde{F}$ and starts at x from B.

Case 3.2.2. f is not incident with x.

Let ${P}_{a}$ be a 3-path starting at a, and let it contain f, then a is adjacent to x.

Case 3.2.2.1. $|{F}^{*}|=0$.

Here, $|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=1$. Without loss of generality, let $|{F}_{e}^{1}|=1$. Now we consider $a=\left(1i\right)$ for $2\le i\le n-1$. Without loss of generality, let $a=\left(12\right)$. We can choose a 2-path ${P}_{{a}^{{}^{\prime}}}=\langle {a}^{{}^{\prime}},{a}^{{}^{\prime}}\left(23\right),{a}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{2}$, where ${a}^{{}^{\prime}}=\left(12\right)\left(1n\right)$. Notice that $|{F}_{e}^{2}|=0$, so ${P}_{{a}^{{}^{\prime}}}$ does not contain a missing edge. By Claim 1 and $|{F}_{e}^{i}|\le 1$ for $i=1,2,n-1$, we can find $(2n-6)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{i}$, and any two of these 3-paths have no common node except u (resp. v, w). Notice that $v\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$, $|V\left({P}_{{a}^{{}^{\prime}}}\right)\cap V\left(C{W}_{n}^{2}\right)|=3$ and $2n-6>3$ for $n\ge 6$, so we can find a 3-path ${P}_{v}$ that does not contain a missing edge in $C{W}_{n}^{2}$, and ${P}_{v}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node. We can find a 3-path ${P}_{u}$ (resp. ${P}_{w}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{n-1}$) as well. We can connect ${P}_{{a}^{{}^{\prime}}}$ to a to obtain ${P}_{a}$. Notice that each of ${P}_{a}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ does not contain a missing edge, and any two of them have no common node. We connect ${P}_{a}$, ${P}_{u}$, ${P}_{v}$, ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ can be got in $C{W}_{n}-{F}_{e}$. The case of $a=(i,i+1)$ for $2\le i\le n-2$ can be proved similarly.

Case 3.2.2.2. $|{F}^{*}|=1$.

Here, $|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=0$. Now we consider $a=\left(1i\right)$ for $2\le i\le n-1$. Without loss of generality, let $a=\left(12\right)$. We can find ${P}_{{a}_{1}^{{}^{\prime}}}=\langle {a}_{1}^{{}^{\prime}},{a}_{1}^{{}^{\prime}}\left(23\right),{a}_{1}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{2}$ and ${P}_{{a}_{2}^{{}^{\prime}}}=\langle {a}_{2}^{{}^{\prime}},{a}_{2}^{{}^{\prime}}\left(23\right),{a}_{2}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{1}$, where ${a}_{1}^{{}^{\prime}}=\left(12\right)\left(1n\right)$ and ${a}_{2}^{{}^{\prime}}=\left(12\right)\left(2n\right)$. Since $|{F}_{e}^{1}|=|{F}_{e}^{2}|=0$, we can choose a 2-path ${P}_{{a}^{{}^{\prime}}}$ from ${P}_{{a}_{1}^{{}^{\prime}}}$ and ${P}_{{a}_{2}^{{}^{\prime}}}$ that does not contain a missing edge. Notice that ${P}_{{a}_{1}^{{}^{\prime}}}$ and ${P}_{{a}_{2}^{{}^{\prime}}}$ are in different $C{W}_{n}^{i}$’s and $|{F}^{*}|=1$, and we can connect ${P}_{{a}^{{}^{\prime}}}$ to a to obtain a 3-path ${P}_{a}$ that does not contain a missing edge. By Claim 1 and $|{F}_{e}^{i}|=0$ for $i=1,2,n-1$, we can find $(2n-5)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{i}$, and any two of these 3-paths have no common node except u (resp. v, w). Notice that $u\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$, $v\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$ and $2n-5>3$ for $n\ge 6$. If ${P}_{{a}^{{}^{\prime}}}={P}_{{a}_{2}^{{}^{\prime}}}$, then we can find a 3-path ${P}_{u}$ that does not contain a missing edge in $C{W}_{n}^{1}$, and ${P}_{u}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node, and we can find a 3-path ${P}_{v}$ (resp. ${P}_{w}$) that does not contain a missing edge in $C{W}_{n}^{2}$ (resp. $C{W}_{n}^{n-1}$) as well. If ${P}_{{a}^{{}^{\prime}}}={P}_{{a}_{1}^{{}^{\prime}}}$, then we can find a 3-path ${P}_{v}$ that does not contain a missing edge in $C{W}_{n}^{2}$, and ${P}_{v}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node, and we can find a 3-path ${P}_{u}$ (resp. ${P}_{w}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{n-1}$) as well. We connect ${P}_{a}$, ${P}_{u}$, ${P}_{v}$, ${P}_{w}$ to x. Notice that each of ${P}_{a}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ does not contain a missing edge, and any two of them have no common node. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|=1$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any link of $\tilde{F}$ and starts at x from B. The case of $a=(i,i+1)$ for $2\le i\le n-2$ can be proved similarly.

Case 4. $|{F}_{e}^{n}|=2n-6,|{F}_{e}|=2n-4$.

Here, $|{F}^{*}|+|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=2$. By Claim 1, we can choose a 3-path ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) from at least $(2n-7)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$, $C{W}_{n}^{n-1}$). Let f be an arbitrary link of ${F}_{e}^{n}$, and let ${F}_{e}^{{}^{\prime}}={F}_{e}^{n}\backslash \left\{f\right\}$. Then ${F}_{e}^{{}^{\prime}}=2n-7$, so there exists an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$ in $C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}$ by Claim 1. Let ${A}^{{}^{\prime}}=ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$.

Case 4.1. $f\notin E\left({A}^{{}^{\prime}}\right)$.

Here, ${A}^{{}^{\prime}}$ is one of extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{n}}\left(x\right))$ at x in $C{W}_{n}^{n}-{F}_{e}^{n}$. We connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with ${A}^{{}^{\prime}}$, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|\le 2$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any link of $\tilde{F}$ and starts at x from B.

Case 4.2. $f\in E\left({A}^{{}^{\prime}}\right)$.

Let ${P}_{x}$ be a 4-path containing f and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ from ${A}^{{}^{\prime}}$, denoted by A.

Case 4.2.1. f is incident with x.

Connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, Then, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|\le 2$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any link of $\tilde{F}$ and starts at x from B.

Case 4.2.2. f is not incident with x.

Let ${P}_{a}$ be a 3-path starting at a and containing f, then a is adjacent to x.

Case 4.2.2.1. $|{F}^{*}|=0$.

Here, $|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=2$.

Case 4.2.2.1.1. $|{F}_{e}^{i}|=2$ for some $i\in \{1,2,\cdots ,n-1\}$.

Without loss of generality, let $|{F}_{e}^{1}|=2$. We can complete the proof as Case 3.2.2.1.

Case 4.2.2.1.2. $|{F}_{e}^{i}|=1$ and $|{F}_{e}^{j}|=1$ for some $i,j\in \{1,2,\cdots ,n-1\}$, $i\ne j$.

Without loss of generality, let $|{F}_{e}^{1}|=1$ and $|{F}_{e}^{2}|=1$. Now we consider $a=\left(1i\right)$ for $2\le i\le n-1$. Without loss of generality, let $a=\left(12\right)$. We can choose a 2-path ${P}_{{a}^{{}^{\prime}}}=\langle {a}^{{}^{\prime}},{a}^{{}^{\prime}}\left(23\right),{a}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{n-1}$, where ${a}^{{}^{\prime}}=\left(12\right)(n-1,n)$. Notice that $|{F}_{e}^{n-1}|=0$, so ${P}_{{a}^{{}^{\prime}}}$ does not contain a missing edge. By Claim 1 and $|{F}_{e}^{i}|\le 1$ for $i=1,2,n-1$, we can find $(2n-6)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{i}$, and any two of these 3-paths have no common node except u (resp. v, w). Notice that $w\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$, and $2n-6>3$ for $n\ge 6$, so we can find a 3-path ${P}_{w}$ that does not contain a missing edge in $C{W}_{n}^{n-1}$, and ${P}_{w}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node. We can find a 3-path ${P}_{u}$ (resp. ${P}_{v}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$). Connect ${P}_{{a}^{{}^{\prime}}}$ to a to obtain a 3-path ${P}_{a}$. Notice that each of ${P}_{a}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ does not contain a missing edge, and any two of them have no common node. We connect ${P}_{a}$, ${P}_{u}$, ${P}_{v}$, ${P}_{w}$ to x. Combining them with A, we can get the extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$. The case of $a=(i,i+1)$ for $2\le i\le n-2$ can be proved similarly.

Case 4.2.2.2. $|{F}^{*}|=1$.

Here, $|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=1$. Without loss of generality, let $|{F}_{e}^{1}|=1$. Next, we consider $a=\left(1i\right)$ for $2\le i\le n-1$. Without loss of generality, let $a=\left(12\right)$. We can find ${P}_{{a}_{1}^{{}^{\prime}}}=\langle {a}_{1}^{{}^{\prime}},{a}_{1}^{{}^{\prime}}\left(23\right),{a}_{1}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{2}$ and ${P}_{{a}_{2}^{{}^{\prime}}}=\langle {a}_{2}^{{}^{\prime}},{a}_{2}^{{}^{\prime}}\left(23\right),{a}_{2}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{n-1}$, where ${a}_{1}^{{}^{\prime}}=\left(12\right)\left(1n\right)$ and ${a}_{2}^{{}^{\prime}}=\left(12\right)(n-1,n)$. Since $|{F}_{e}^{2}|=|{F}_{e}^{n-1}|=0$, ${P}_{{a}_{1}^{{}^{\prime}}}$ and ${P}_{{a}_{2}^{{}^{\prime}}}$ do not contain a missing edge. We can choose a 2-path ${P}_{{a}^{{}^{\prime}}}$ from ${P}_{{a}_{1}^{{}^{\prime}}}$ and ${P}_{{a}_{2}^{{}^{\prime}}}$. Since ${P}_{{a}_{1}^{{}^{\prime}}}$ and ${P}_{{a}_{2}^{{}^{\prime}}}$ are in different $C{W}_{n}^{i}$’s and $|{F}^{*}|=1$, connect ${P}_{{a}^{{}^{\prime}}}$ to a to obtain a 3-path ${P}_{a}$ that does not contain a missing edge. By Claim 1 and $|{F}_{e}^{i}|\le 1$ for $i=1,2,n-1$, we can find $(2n-6)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{i}$, and any two of these 3-paths have no common node except u (resp. v, w). Notice that $v\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$, $w\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$ and $2n-6>3$ for $n\ge 6$. If ${P}_{{a}^{{}^{\prime}}}={P}_{{a}_{1}^{{}^{\prime}}}$, then we can find a 3-path ${P}_{v}$ that does not contain a missing edge in $C{W}_{n}^{2}$, and ${P}_{v}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node, and we can find a 3-path ${P}_{u}$ (resp. ${P}_{w}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{n-1}$) as well. If ${P}_{{a}^{{}^{\prime}}}={P}_{{a}_{2}^{{}^{\prime}}}$, then we can find a 3-path ${P}_{w}$ that does not contain a missing edge in $C{W}_{n}^{n-1}$, and ${P}_{w}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node, and we can find a 3-path ${P}_{u}$ (resp. ${P}_{v}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$) as well. We connect ${P}_{a}$, ${P}_{u}$, ${P}_{v}$, ${P}_{w}$ to x. Notice that each of ${P}_{a}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ does not contain a missing edge, and any two of them have no common node. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|=1$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any edge of $\tilde{F}$ and starts at x from B. The case of $a=(i,i+1)$ for $2\le i\le n-2$ can be proved similarly.

Case 4.2.2.3. $|{F}^{*}|=2$.

Here, $|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=0$. Now we consider $a=\left(1i\right)$ for $2\le i\le n-1$. Without loss of generality, let $a=\left(12\right)$. We can find ${P}_{{a}_{1}^{{}^{\prime}}}=\langle {a}_{1}^{{}^{\prime}},{a}_{1}^{{}^{\prime}}\left(23\right),{a}_{1}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{1}$, ${P}_{{a}_{2}^{{}^{\prime}}}=\langle {a}_{2}^{{}^{\prime}},{a}_{2}^{{}^{\prime}}\left(23\right),{a}_{2}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{2}$ and ${P}_{{a}_{3}^{{}^{\prime}}}=\langle {a}_{3}^{{}^{\prime}},{a}_{3}^{{}^{\prime}}\left(23\right),{a}_{3}^{{}^{\prime}}\left(23\right)\left(34\right)\rangle $ in $C{W}_{n}^{n-1}$, where ${a}_{1}^{{}^{\prime}}=\left(12\right)\left(2n\right)$, ${a}_{2}^{{}^{\prime}}=\left(12\right)\left(1n\right)$ and ${a}_{3}^{{}^{\prime}}=\left(12\right)(n-1,n)$. Since $|{F}_{e}^{1}|=|{F}_{e}^{2}|=|{F}_{e}^{n-1}|=0$, then ${P}_{{a}_{1}^{{}^{\prime}}}$, ${P}_{{a}_{2}^{{}^{\prime}}}$ and ${P}_{{a}_{3}^{{}^{\prime}}}$ do not contain a missing edge. Notice that ${P}_{{a}_{1}^{{}^{\prime}}}$, ${P}_{{a}_{2}^{{}^{\prime}}}$ and ${P}_{{a}_{3}^{{}^{\prime}}}$ are in different $C{W}_{n}^{i}$’s and $|{F}^{*}|=2$, then we can choose a 2-path ${P}_{{a}^{{}^{\prime}}}$ from ${P}_{{a}_{1}^{{}^{\prime}}}$, ${P}_{{a}_{2}^{{}^{\prime}}}$ and ${P}_{{a}_{3}^{{}^{\prime}}}$, and attach the 2-path ${P}_{{a}^{{}^{\prime}}}$ to a to obtain a 3-path ${P}_{a}$ that does not contain a missing edge. By Claim 1 and $|{F}_{e}^{i}|=0$ for $i=1,2,n-1$, we can find $(2n-5)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{i}$, and any two of these 3-paths have no common node except u (resp. v, w). Notice that $u,v,w\notin V\left({P}_{{a}^{{}^{\prime}}}\right)$, and $2n-5>3$ for $n\ge 6$. If ${P}_{{a}^{{}^{\prime}}}={P}_{{a}_{1}^{{}^{\prime}}}$, then we can find a 3-path ${P}_{u}$ that does not contain a missing edge in $C{W}_{n}^{1}$ such that ${P}_{u}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node, and we can find a 3-path ${P}_{v}$ (resp. ${P}_{w}$) that does not contain a missing edge in $C{W}_{n}^{2}$ (resp. $C{W}_{n}^{n-1}$) as well. If ${P}_{{a}^{{}^{\prime}}}={P}_{{a}_{2}^{{}^{\prime}}}$, then we can find a 3-path ${P}_{v}$ in $C{W}_{n}^{2}$ that does not contain a missing edge, and ${P}_{v}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node, and we can find a 3-path ${P}_{u}$ (resp. ${P}_{w}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{n-1}$) as well. If ${P}_{{a}^{{}^{\prime}}}={P}_{{a}_{3}^{{}^{\prime}}}$, then we can find a 3-path ${P}_{w}$ in $C{W}_{n}^{n-1}$ that does not contain a missing edge, and ${P}_{w}$ and ${P}_{{a}^{{}^{\prime}}}$ have no common node, and we can find a 3-path ${P}_{u}$ (resp. ${P}_{v}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$). Notice that each of ${P}_{a}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ does not contain a missing edge, and any two of them have no common node. We connect ${P}_{a}$, ${P}_{u}$, ${P}_{v}$, ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|=2$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any link of $\tilde{F}$ and starts at x from B. The case of $a=(i,i+1)$ for $2\le i\le n-2$ can be proved similarly.

Case 5. $|{F}_{e}^{n}|=2n-5,|{F}_{e}|=2n-5$.

Here, $|{F}^{*}|+|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=0$. By Claim 1, we can choose a 3-path ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) from at least $(2n-5)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$, $C{W}_{n}^{n-1}$). Let $f,{f}^{{}^{\prime}}$ be any two elements of ${F}_{e}^{n}$ and ${F}_{e}^{{}^{\prime}}={F}_{e}^{n}\backslash \{f,{f}^{{}^{\prime}}\}$. Then $|{F}_{e}^{{}^{\prime}}|=2n-7$, so there exists an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$ in $C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}$ by Claim 1. Let ${A}^{{}^{\prime}}=ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$.

Case 5.1. Neither f nor $f{}^{\prime}$ belongs to ${A}^{{}^{\prime}}$.

Here, ${A}^{{}^{\prime}}$ is one of extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{n}}\left(x\right))$ at x in $C{W}_{n}^{n}-{F}_{e}^{n}$. Notice that $|{F}^{*}|=0$, so we connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with ${A}^{{}^{\prime}}$, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$.

Case 5.2. ${A}^{{}^{\prime}}$ contains f or $f{}^{\prime}$.

Without loss of generality, we assume that ${A}^{{}^{\prime}}$ contains only f. Let ${P}_{x}$ be a 4-path containing f and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ from ${A}^{{}^{\prime}}$, denoted by A. Next we discuss whether f is incident with x or not. If f is incident with x, then it can be proved as Case 2.2.1. If f is not incident with x, then it can be proved as Case 2.2.2.

Case 5.3. ${A}^{{}^{\prime}}$ contains f and $f{}^{\prime}$.

Case 5.3.1. f and ${f}^{{}^{\prime}}$ are both incident with x.

Let ${P}_{x}$ (resp. ${P}_{{x}^{{}^{\prime}}}$) be a 4-path containing f (resp. ${f}^{{}^{\prime}}$) and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ and ${P}_{{x}^{{}^{\prime}}}$ from ${A}^{{}^{\prime}}$, denoted by A. Notice that $|{F}^{*}|=0$, so we connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$.

Case 5.3.2. Just one of f and ${f}^{{}^{\prime}}$ is incident with x.

Without loss of generality, assumes that only ${f}^{{}^{\prime}}$ is incident with x. Let ${P}_{x}$ (resp. ${P}_{{x}^{{}^{\prime}}}$) be a 4-path containing f (resp. ${f}^{{}^{\prime}}$) and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ and ${P}_{{x}^{{}^{\prime}}}$ from ${A}^{{}^{\prime}}$, denoted by A. Let ${P}_{a}$ be a 3-path containing f and starting at a, and then a is adjacent to x. The next proof can be completed as Case 2.2.2.

Case 5.3.3. Neither f nor ${f}^{{}^{\prime}}$ is incident with x.

Next we discuss whether f and ${f}^{{}^{\prime}}$ belong to the same path or not.

Case 5.3.3.1. f and ${f}^{{}^{\prime}}$ belong to the same path in ${A}^{{}^{\prime}}$.

Then we can complete the proof as Case 2.2.2.

Case 5.3.3.2. f and ${f}^{{}^{\prime}}$ belong to different paths in ${A}^{{}^{\prime}}$.

Let ${P}_{a}$ (resp. ${P}_{b}$) be a 3-path starting at a (resp. b), and it contains f (resp. ${f}^{{}^{\prime}}$). Then a and b are both incident with x. Notice that $|{F}_{e}^{n}|=2n-5,|{F}_{e}|=2n-5$. It is easy to find a 3-path ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$) that does not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$), and ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$) and ${P}_{u}$ (resp. ${P}_{v}$) have no common vertices. Connecting ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$) to a (resp. b), we can obtain a 3-path ${P}_{a}$ (resp. ${P}_{b}$). Notice that $|{F}^{*}|=0$, so we connect ${P}_{a}$, ${P}_{b}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$.

Case 6. $|{F}_{e}^{n}|=2n-5,|{F}_{e}|=2n-4$.

Here, $|{F}^{*}|+|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=1$. By Claim 1, we can choose a 3-path ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) from at least $(2n-6)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$, $C{W}_{n}^{n-1}$). Let $f,{f}^{{}^{\prime}}$ be any two elements of ${F}_{e}^{n}$ and ${F}_{e}^{{}^{\prime}}={F}_{e}^{n}\backslash \{f,{f}^{{}^{\prime}}\}$. Then $|{F}_{e}^{{}^{\prime}}|=2n-7$, there exists an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$ in $C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}$ by Claim 1. Let ${A}^{{}^{\prime}}=ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$.

Case 6.1. Neither f nor ${f}^{{}^{\prime}}$ belongs to ${A}^{{}^{\prime}}$.

Here, ${A}^{{}^{\prime}}$ is one of extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{n}}\left(x\right))$ at x in $C{W}_{n}^{n}-{F}_{e}^{n}$. Connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with ${A}^{{}^{\prime}}$, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ is found in $C{W}_{n}-{F}_{e}$ at x, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|\le 1$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any link of $\tilde{F}$ and starts at x from B.

Case 6.2. ${A}^{{}^{\prime}}$ contains f or ${f}^{{}^{\prime}}$.

Without loss of generality, we suppose that ${A}^{{}^{\prime}}$ contains only f. Let ${P}_{x}$ be a 4-path containing f and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ from ${A}^{{}^{\prime}}$, denoted by A. Next we consider whether f is incident with x or not. If f is incident with x, then the next proof can be completed as Case 3.2.1. If f is not incident with x, then the next proof can be completed as Case 3.2.2.

Case 6.3. ${A}^{{}^{\prime}}$ contains f and ${f}^{{}^{\prime}}$.

Case 6.3.1. f and ${f}^{{}^{\prime}}$ are both incident with x.

Let ${P}_{x}$ (resp. ${P}_{{x}^{{}^{\prime}}}$) be a 4-path containing f (resp. ${f}^{{}^{\prime}}$) and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ and ${P}_{{x}^{{}^{\prime}}}$ from ${A}^{{}^{\prime}}$, denoted by A. Connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ at x is found in $C{W}_{n}-{F}_{e}$, (with $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ written simply as B). Notice that $|{F}^{*}|\le 1$. If $\tilde{F}=\varnothing $, then B satisfies the lemma. If $\tilde{F}\ne \varnothing $, then we find an extended star to satisfy the lemma by removing any path that contains any link of $\tilde{F}$ and starts at x from B.

Case 6.3.2. Just one of f and ${f}^{{}^{\prime}}$ is incident with x.

Without loss of generality, assume that only ${f}^{{}^{\prime}}$ is incident with x. Let ${P}_{x}$ (resp. ${P}_{{x}^{{}^{\prime}}}$) be a 4-path containing f (resp. ${f}^{{}^{\prime}}$) and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$ and ${P}_{{x}^{{}^{\prime}}}$ from ${A}^{{}^{\prime}}$, denoted by A. Let ${P}_{a}$ be a 3-path containing f and starting at a, and then a is adjacent to x. We can complete the proof as in Case 3.2.2.

Case 6.3.3. Neither f nor ${f}^{{}^{\prime}}$ is incident with x.

Then we consider whether f and ${f}^{{}^{\prime}}$ belong to the same path or not.

Case 6.3.3.1. f and ${f}^{{}^{\prime}}$ belong to the same path in ${A}^{{}^{\prime}}$.

Then we can complete the proof as in Case 3.2.2.

Case 6.3.3.2. f and ${f}^{{}^{\prime}}$ belong to different paths in ${A}^{{}^{\prime}}$.

Let ${P}_{a}$ (resp. ${P}_{b}$) be a 3-path starting at a (resp. b), and it contains f (resp. ${f}^{{}^{\prime}}$). Then a and b are both incident with x. Notice that $|{F}_{e}^{n}|=2n-5,|{F}_{e}|=2n-4$, then $|{F}^{*}|\le 1$. Note that $|{F}_{e}^{i}|\le 1$, $i=1,2,n-1$. There is a 2-path ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$) that do not contain a missing edge in $C{W}_{n}^{1}$, or $C{W}_{n}^{2}$ or $C{W}_{n}^{n-1}$, and the edge $a{a}^{{}^{\prime}}\notin F$ (resp. $b{b}^{{}^{\prime}}\notin F$), where ${a}^{{}^{\prime}}$ (resp. ${b}^{{}^{\prime}}$) is an outside neighbor of a (resp. b). Connecting ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$) to a (resp. b), we can obtain a 3-path ${P}_{a}$ (resp. ${P}_{b}$). Connect ${P}_{a}$, ${P}_{b}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$.

Case 7. $|{F}_{e}^{n}|=2n-4,|{F}_{e}|=2n-4$.

Here, $|{F}^{*}|+|{F}_{e}^{1}|+|{F}_{e}^{2}|+\cdots +|{F}_{e}^{n-1}|=0$. By Claim 1, we can choose a 3-path ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) from at least $(2n-5)$ 3-paths that do not contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$, $C{W}_{n}^{n-1}$). Let $f,{f}^{{}^{\prime}},{f}^{{}^{\u2033}}$ be any three elements of ${F}_{e}^{n}$ and ${F}_{e}^{{}^{\prime}}={F}_{e}^{n}\backslash \{f,{f}^{{}^{\prime}},{f}^{{}^{\u2033}}\}$. Then $|{F}_{e}^{{}^{\prime}}|=2n-7$, by Claim 1, there exists an extended star $ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$ in $C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}$. Let ${A}^{{}^{\prime}}=ES(x;{d}_{C{W}_{n}^{n}-{F}_{e}^{{}^{\prime}}}\left(x\right))$.

Case 7.1. None of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ belongs to ${A}^{{}^{\prime}}$.

We can complete the proof as in Case 5.1.

Case 7.2. ${A}^{{}^{\prime}}$ contains just one of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$.

We can complete the proof as in Case 5.2.

Case 7.3. ${A}^{{}^{\prime}}$ contains just two of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$.

We can complete the proof as in Case 5.3.

Case 7.4. ${A}^{{}^{\prime}}$ contains f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$.

Case 7.4.1. f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ are all incident with x.

Let ${P}_{x}$ (resp. ${P}_{{x}^{{}^{\prime}}}$, ${P}_{{x}^{{}^{\u2033}}}$) be a 4-path containing f (resp. ${f}^{{}^{\prime}}$, ${f}^{{}^{\u2033}}$) and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$, ${P}_{{x}^{{}^{\prime}}}$ and ${P}_{{x}^{{}^{\u2033}}}$ from ${A}^{{}^{\prime}}$, denoted by A. Notice that $|{F}^{*}|=0$, so we connect ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with A, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$.

Case 7.4.2. Just two of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ are incident with x.

Without loss of generality, assumes that ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ are incident with x. Let ${P}_{x}$ (resp. ${P}_{{x}^{{}^{\prime}}}$, ${P}_{{x}^{{}^{\u2033}}}$) be a 4-path containing f (resp. ${f}^{{}^{\prime}}$, ${f}^{{}^{\u2033}}$) and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$, ${P}_{{x}^{{}^{\prime}}}$ and ${P}_{{x}^{{}^{\u2033}}}$ from ${A}^{{}^{\prime}}$, denoted by A. Let ${P}_{a}$ be a 3-path containing f and starting at a, and then a is adjacent to x. The next proof can be completed as in Case 5.3.2.

Case 7.4.3. Just one of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ is incident with x.

Without loss of generality, assumes that only ${f}^{{}^{\prime}}$ is incident with x. Let ${P}_{x}$ (resp. ${P}_{{x}^{{}^{\prime}}}$, ${P}_{{x}^{{}^{\u2033}}}$) be a 4-path containing f (resp. ${f}^{{}^{\prime}}$, ${f}^{{}^{\u2033}}$) and starting at x in ${A}^{{}^{\prime}}$. A graph is obtained by removing ${P}_{x}$, ${P}_{{x}^{{}^{\prime}}}$ and ${P}_{{x}^{{}^{\u2033}}}$ from ${A}^{{}^{\prime}}$, denoted by A. Let ${P}_{a}$ be a 3-path containing f and starting at a, and then a is adjacent to x. The next proof can be completed as in Case 5.3.3.

Case 7.4.4. None of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ is incident with x.

Then we consider whether f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ belong to the same path or not.

Case 7.4.4.1. f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ belong to the same path in ${A}^{{}^{\prime}}$.

Then we can complete the proof as in Case 2.2.2.

Case 7.4.4.2. f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ belong to different paths in ${A}^{{}^{\prime}}$.

Case 7.4.4.2.1. Just two of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ belong to a path in ${A}^{{}^{\prime}}$.

Without loss of generality, assumes ${P}_{a}$ is a 3-path containing f and ${f}^{{}^{\u2033}}$ and starting at a, ${P}_{b}$ is a 3-path containing ${f}^{{}^{\prime}}$ and starting at b. Then a and b are both incident with x. Then we can complete the proof as in Case 5.3.3.

Case 7.4.4.2.2. Each of f, ${f}^{{}^{\prime}}$ and ${f}^{{}^{\u2033}}$ belong to a path in ${A}^{{}^{\prime}}$ separately. Let ${P}_{a}$ (resp. ${P}_{b}$, ${P}_{c}$) be a 3-path starting at a (resp. b, c), and it contains f (resp. ${f}^{{}^{\prime}}$, ${f}^{{}^{\u2033}}$). Then a, b and c are all incident with x. Notice that $|{F}_{e}^{n}|=2n-4,|{F}_{e}|=2n-4$. It is easy to find a 3-path ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$, ${P}_{{c}^{{}^{\prime}}}$) that does contain a missing edge in $C{W}_{n}^{1}$ (resp. $C{W}_{n}^{2}$, $C{W}_{n}^{n-1}$), and ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$, ${P}_{{c}^{{}^{\prime}}}$) and ${P}_{u}$ (resp. ${P}_{v}$, ${P}_{w}$) have no common vertices. Connecting ${P}_{{a}^{{}^{\prime}}}$ (resp. ${P}_{{b}^{{}^{\prime}}}$, ${P}_{{c}^{{}^{\prime}}}$) to a (resp. b, c), we can obtain a 3-path ${P}_{a}$ (resp. ${P}_{b}$, ${P}_{c}$). Notice that $|{F}^{*}|=0$, so we connect ${P}_{a}$, ${P}_{b}$, ${P}_{c}$, ${P}_{u}$, ${P}_{v}$ and ${P}_{w}$ to x. Combining them with ${A}^{{}^{\prime}}$, we can get an extended star $ES(x;{d}_{C{W}_{n}-{F}_{e}}\left(x\right))$ in $C{W}_{n}-{F}_{e}$. □