Hamilton-connectivity of line graphs with application to their detour index

Abstract

A graph is called Hamilton-connected if there exists a Hamiltonian path between every pair of its vertices. Determining whether or not a graph is Hamilton-connected is an NP-complete problem. In this paper, we present two methods to show Hamilton-connectivity in graphs. The first method uses the vertex connectivity and Hamiltoniancity of graphs, and, the second is the definition-based constructive method which constructs Hamiltonian paths between every pair of vertices. By employing these proof techniques, we show that the line graphs of the generalized Petersen, anti-prism and wheel graphs are Hamilton-connected. Combining it with some existing results, it shows that some of these families of Hamilton-connected line graphs have their underlying graph families non-Hamilton-connected. This, in turn, shows that the underlying graph of a Hamilton-connected line graph is not necessarily Hamilton-connected. As side results, the detour index being also an NP-complete problem, has been calculated for the families of Hamilton-connected line graphs. Finally, by computer we generate all the Hamilton-connected graphs on $\le 7$ vertices and all the Hamilton-laceable graphs on $\le 10$ vertices. Our research contributes towards our proposed conjecture that almost all graphs are Hamilton-connected.

Appendices

Appendix: Remaining proof of Theorem 7

Proof

Case 2: If $x=u_1$ and $y=w_i$

Subcase 2.1: $i=1$

$$\begin{aligned} P_H(x,y):x=u_1u_nv_n\sim \{v_j : 1\le j\le n \}\sim \{w_{n-j}u_{n-j} : 1\le j\le n-2 \}\sim w_{1}=y \end{aligned}$$

Subcase 2.2: $i\ge 2,~n\not \mid 2,~i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-1 \}\sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim v_iv_i =y \end{aligned}$$

Subcase 2.3: $i\ge 2,~n\not \mid 2,~i\mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \}\sim w_{i-1}v_{i}v_{i-1}\\&\quad \sim \{w_{i-j}v_{i-j} : 2\le j\le i-1 \}\sim v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_i =y \end{aligned}$$

Subcase 2.4: $i\ge 2,~n\mid 2,~i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \}\sim w_{i-1}v_{i}v_{i-1}\\&\quad \sim \{w_{i-j}v_{i-j} : 2\le j\le i-1 \}\sim v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_i =y \end{aligned}$$

Subcase 2.5: $i\ge 2,~n\mid 2,~i\mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-1 \}\sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim v_iv_i =y \end{aligned}$$

Case 3: If $x=u_1$ and $y=v_i$

Subcase 3.1: $i=1$

$$\begin{aligned} P_H(x,y):x=u_1\sim \{u_j : 2\le j\le n \}\sim \{w_{n-j}v_{n-j} : 0\le j\le n-2 \}\sim w_{1}v_{1}=y \end{aligned}$$

Subcase 3.2: $i\ge 2,~ n\not \mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \}\\&\quad \sim \{w_{i-j}v_{i-j} : 1\le j\le i-1 \}\sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 3.3: $i\ge 2,~ n\not \mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \} \sim \{w_{i-j}v_{i-j} :1\le j\le i-1 \}\sim v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 3.4: $i\ge 2,~ n\mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \}\sim w_{i-1}\\&\quad \sim \{w_{i-j}v_{i-j} :1\le j\le i-1 \}\sim v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 3.5: $i\ge 2,~ n\mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=u_1\sim \{u_{j} : 2\le j\le i \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-1 \}\sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Case 4: $x=w_1$ and $y=w_i, ~ i\ne 1$

Subcase 4.1: $i=2$

$$\begin{aligned}&P_H(x,y):x=w_1u_1\sim \{u_j : 2\le j\le n \}\sim w_{n}v_{1}v_{n}\\&\quad \sim \{w_{n-j}v_{n-j} : 1\le j\le n-3 \}\sim v_{2}w_{2}=y \end{aligned}$$

Subcase 4.2: $i\ge 3, n\not \mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-1 \}\sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim v_iv_i =y \end{aligned}$$

Subcase 4.3: $i\ge 3, n\not \mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \}\sim w_{i-1}v_{i}v_{i-1}\\&\quad \sim \{w_{i-j}v_{i-j} :2\le j\le i-1 \}\sim v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_i =y \end{aligned}$$

Subcase 4.4: $i\ge 3, n\mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \}\sim w_{i-1}v_{i}v_{i-1}\\&\quad \sim \{w_{i-j}v_{i-j} :2\le j\le i-1 \}\sim v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_i =y \end{aligned}$$

Subcase 4.5: $i\ge 3, n\mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-1 \}\sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim v_iv_i =y \end{aligned}$$

Case 5: $x=w_1$ and $y=u_i$

Subcase 5.1: $i=1$

$$\begin{aligned} P_H(x,y):x=w_1u_1\sim \{v_j : 1\le j\le n \} \sim \{w_{n-j}u_{n-j} : 0\le j\le n-2 \}\sim u_{1}=y \end{aligned}$$

Subcase 5.2: $i\ge 2,~ n\not \mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i-1 \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-2 \}\\&\quad \sim v_1 \sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim v_iv_iw_i =y \end{aligned}$$

Subcase 5.3: $i\ge 2,~ n\not \mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i-1 \} \sim w_{i-1}v_{i}v_{i-1}\\&\quad \sim \{w_{i-j}v_{i-j} :2\le j\le i-2 \}\sim v_{1} v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iw_i =y \end{aligned}$$

Subcase 5.4: $i\ge 2,~ n\mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i-1 \} \sim w_{i-1}v_{i}v_{i-1}\\&\quad \sim \{w_{i-j}v_{i-j} :2\le j\le i-2 \}\sim v_{1} v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iw_i =y \end{aligned}$$

Subcase 5.5: $i\ge 2,~ n\mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i-1 \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-2 \}\\&\quad \sim v_1 \sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim v_iv_iw_i =y \end{aligned}$$

Case 6: $x=w_1$ and $y=v_i$

Subcase 6.1: $i=1$

$$\begin{aligned} P_H(x,y):x=w_1u_1\sim \{u_j : 1\le j\le n \} \sim \{w_{n-j}v_{n-j} : 0\le j\le n-2 \}\sim v_{1}=y \end{aligned}$$

Subcase 6.2: $i\ge 2,~ n\not \mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-2 \}\\&\quad \sim v_1 \sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 6.3: $i\ge 2,~ n\not \mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \} \sim \{w_{i-j}v_{i-j} :1\le j\le i-2 \}\sim v_{1} v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 6.4: $i\ge 2,~ n\mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \} \sim \{w_{i-j}v_{i-j} :1\le j\le i-2 \}\sim v_{1} v_{n}w_{n}u_{n} \\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j}v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 6.5: $i\ge 2,~ n\mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{u_{j} : 1\le j\le i \}\sim \{w_{i-j}v_{i-j} : 1\le j\le i-2 \}\\&\quad \sim v_1 \sim \{v_{n-j}w_{n-j}u_{n-j} \\&\quad u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Case 7: $x=v_1$ and $y=v_i, ~ i\ne 1$

Subcase 7.1: $i=2$

$$\begin{aligned} P_H(x,y):x=v_1w_1u_1\sim \{u_jv_j : 2\le j\le n \} \sim \{v_{n-j} : 0\le j\le n-3 \}\sim v_{2}=y \end{aligned}$$

Subcase 7.2: $i\ge 3, n\not \mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i-1 \}\\&\quad \sim \{w_{i-j}u_{i-j} : 1\le j\le i-1 \}\sim \{u_{n-j}w_{n-j}v_{n-j} \\&\quad v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim u_iv_iu_i =y \end{aligned}$$

Subcase 7.3: $i\ge 3, n\not \mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i-1 \}\sim w_{i-1}u_{i}u_{i-1}\\&\quad \sim \{w_{i-j}u_{i-j} :2\le j\le i-1 \}\sim u_{n}w_{n}v_{n} \\&\quad \sim \{v_{n-j}w_{n-j}u_{n-j}u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 7.4: $i\ge 3, n\mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i-1 \}\sim w_{i-1}u_{i}u_{i-1}\\&\quad \sim \{w_{i-j}u_{i-j} :2\le j\le i-1 \}\sim u_{n}w_{n}v_{n} \\&\quad \sim \{v_{n-j}w_{n-j}u_{n-j}u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iu_i =y \end{aligned}$$

Subcase 7.5: $i\ge 3, n\mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i-1 \}\sim \{w_{i-j}u_{i-j} : 1\le j\le i-1 \}\sim \{u_{n-j}w_{n-j}v_{n-j} \\&\quad v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim u_iv_iu_i =y \end{aligned}$$

Case 8: $x=v_1$ and $y=u_i$

Subcase 8.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v_1\sim \{v_j : 2\le j\le n \} \sim \{w_{n-j}u_{n-j} : 0\le j\le n-2 \}\sim w_1u_1=y \end{aligned}$$

Subcase 8.2: $i\ge 2,~ n\not \mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \}\sim \{w_{i-j}u_{i-j} : 1\le j\le i-1 \}\sim \{u_{n-j}w_{n-j}v_{n-j} \\&\quad v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim w_iw_i =y \end{aligned}$$

Subcase 8.3: $i\ge 2,~ n\not \mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \} \sim \{w_{i-j}u_{i-j} :1\le j\le i-1 \}\sim u_{n}w_{n}v_{n} \\&\quad \sim \{v_{n-j}w_{n-j}u_{n-j}u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iw_i =y \end{aligned}$$

Subcase 8.4: $i\ge 2,~ n\mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \} \sim \{w_{i-j}u_{i-j} :1\le j\le i-1 \}\sim u_{n}w_{n}v_{n} \\&\quad \sim \{v_{n-j}w_{n-j}u_{n-j}u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_iw_i =y \end{aligned}$$

Subcase 8.5: $i\ge 2,~ n\mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \}\sim \{w_{i-j}u_{i-j} : 1\le j\le i-1 \}\sim \{u_{n-j}w_{n-j}v_{n-j} \\&\quad v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim w_iw_i =y \end{aligned}$$

Case 9: $x=v_1$ and $y=w_i$

Subcase 9.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v_1\sim \{v_j : 2\le j\le n \} \sim \{w_{n-j}u_{n-j} : 0\le j\le n-2 \}\sim u_1w_1=y \end{aligned}$$

Subcase 9.2: $i\ge 2,~ n\not \mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \}\sim \{w_{i-j}u_{i-j} : 1\le j\le i-1 \}\sim \{u_{n-j}w_{n-j}v_{n-j} \\&\quad v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim u_iv_i =y \end{aligned}$$

Subcase 9.3: $i\ge 2,~ n\not \mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \} \sim w_{i-1}u_{i}u_{i-1}\\&\quad \sim \{w_{i-j}u_{i-j} :2\le j\le i-1 \}\sim u_{n}w_{n}v_{n} \\&\quad \sim \{v_{n-j}w_{n-j}u_{n-j}u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_i =y \end{aligned}$$

Subcase 9.4: $i\ge 2,~ n\mid 2,~ i\not \mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \} \sim w_{i-1}u_{i}u_{i-1}\\&\quad \sim \{w_{i-j}u_{i-j} :2\le j\le i-1 \}\sim u_{n}w_{n}v_{n} \\&\quad \sim \{v_{n-j}w_{n-j}u_{n-j}u_{n-j-1}w_{n-j-1}v_{n-j-1}:j=1,3,5,\ldots ,n-i-2 \} \sim w_i =y \end{aligned}$$

Subcase 9.5: $i\ge 2,~ n\mid 2,~ i\mid 2$

$$\begin{aligned}&P_H(x,y):x=v_1\sim \{v_{j} : 2\le j\le i \}\sim \{w_{i-j}u_{i-j} : 1\le j\le i-1 \}\\&\quad \sim \{u_{n-j}w_{n-j}v_{n-j} \\&\quad v_{n-j-1}w_{n-j-1}u_{n-j-1}:j=0,2,4,\ldots ,n-i-2 \} \sim u_iv_i =y \end{aligned}$$

$\square$

Remaining proof of Theorem 8

Proof

Case 2: $x=w_1$ and $y=v_i$

Subcase 2.1: $i=1$

$$\begin{aligned} P_H(x,y):x=w_1 \sim \{ v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}: 0\le j\le n-2 \} \sim v^{\prime }_1u_1 v_{1}=y \end{aligned}$$

Subcase 2.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{w_{j} : 2\le j\le i \}\sim \{v^{\prime }_{i-j}u_{i-j}v_{i-j} : 1\le j\le i-1 \} \sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}:0\le j\le n-i-1 \}\sim v^{\prime }_iu_iv_i=y \end{aligned}$$

Case 3: $x=w_1$ and $y=v^{\prime }_i$

Subcase 3.1: $i=1$

$$\begin{aligned} P_H(x,y):x=w_1 \sim \{w_{n-j} v^{\prime }_{n-j}u_{n-j}v_{n-j}: 0\le j\le n-2 \} \sim u_1 v_{1} v^{\prime }_1=y \end{aligned}$$

Subcase 3.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{w_{j} : 2\le j\le i \}\sim v_iu_i \sim \{v^{\prime }_{i-j}u_{i-j}v_{i-j} : 1\le j\le i-1 \} \sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}:0\le j\le n-i-1 \}\sim v^{\prime }_i=y \end{aligned}$$

Case 4: $x=w_1$ and $y=u_i$

Subcase 4.1: $i=1$

$$\begin{aligned} P_H(x,y):x=w_1v_1v^{\prime }_1 \sim \{v_{j}w_{j} v^{\prime }_{j}: 2\le j\le n \} \sim \{u_{n-j}: 0\le j\le n-2 \}\sim u_1=y \end{aligned}$$

Subcase 4.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=w_1\sim \{w_{j} : 2\le j\le i \}\sim \{v^{\prime }_{i-j}u_{i-j}v_{i-j} : 1\le j\le i-1 \} \sim \\&\quad \{v^{\prime }_{n-j}w_{n-j}v_{n-j}u_{n-j}:0\le j\le n-i-1 \}\sim v^{\prime }_iv_iu_i=y \end{aligned}$$

Case 5: $x=v_1$ and $y=v_i$, $i\ne 1$

Subcase 5.1: $i=2$

$$\begin{aligned} P_H(x,y):x=v_1u_1v^{\prime }_1w_1 \sim \{w_{n-j} v^{\prime }_{n-j}u_{n-j}v_{n-j}: 0\le j\le n-3 \} \sim u_2 v^{\prime }_2 w_2v_2=y \end{aligned}$$

Subcase 5.2: $i\ge 3$

$$\begin{aligned}&P_H(x,y):x=v_1u_1v^{\prime }_1\sim \{v_{j}u_{j}v^{\prime }_{j} : 2\le j\le i-1 \}\sim \{w_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{w_{n-j}v^{\prime }_{n-j}u_{n-j}v_{n-j}:0\le j\le n-i-1 \}\sim w_i v^{\prime }_i u_iv_i=y \end{aligned}$$

Case 6: $x=v_1$ and $y=w_i$

Subcase 6.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v_1u_1v^{\prime }_1 \sim \{u_{j}v_{j}w_{j} v^{\prime }_{j}: 2\le j\le n \} \sim w_1=y \end{aligned}$$

Subcase 6.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=v_1u_1v^{\prime }_1 \sim \{v_{j}u_{j} v^{\prime }_{j}: 2\le j\le i-1 \}\sim u_iv_i \sim \{w_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}:0\le j\le n-i-1 \}\sim v^{\prime }_i w_i=y \end{aligned}$$

Case 7: $x=v_1$ and $y=v^{\prime }_i$

Subcase 7.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v_1 \sim \{w_{j}: 1\le j\le n \} \sim \{v^{\prime }_{n-j}u_{n-j}v_{n-j}:0\le j\le n-2 \} \sim u_1v^{\prime }_1=y \end{aligned}$$

Subcase 7.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=v_1u_1v^{\prime }_1 \sim \{v_{j}u_{j} v^{\prime }_{j}: 2\le j\le i-1 \} \sim \{w_{i-j}: 0 \le j \le i-1 \}\sim \\&\quad \{w_{n-j}v^{\prime }_{n-j}u_{n-j}v_{n-j}: 0\le j\le n-i-1 \}\sim u_iv_i v^{\prime }_i=y \end{aligned}$$

Case 8: $x=v_1$ and $y=u_i$

Subcase 8.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v_1w_1v^{\prime }_1 \sim \{w_{j}v_{j}v^{\prime }_j: 2\le j\le n \} \sim \{u_{n-j}:0\le j\le n-2 \} \sim u_1=y \end{aligned}$$

Subcase 8.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=v_1w_1v^{\prime }_1 \sim \{v_{j}w_{j} v^{\prime }_{j}: 2\le j\le i-1 \} \sim \{u_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{u_{n-j}v^{\prime }_{n-j}w_{n-j}v_{n-j}: 0\le j\le n-i-1 \}\sim w_i v^{\prime }_iv_iu_i=y \end{aligned}$$

Case 9: $x=v^{\prime }_1$ and $y=v^{\prime }_i$, $i\ne 1$

Subcase 9.1: $i=2$

$$\begin{aligned} P_H(x,y):x=v^{\prime }_1u_1v_1w_1 \sim \{w_{n-j} v^{\prime }_{n-j}u_{n-j}v_{n-j}: 0\le j\le n-3 \} \sim u_2v_2w_2 v^{\prime }_2=y \end{aligned}$$

Subcase 9.2: $i\ge 3$

$$\begin{aligned}&P_H(x,y):x=v^{\prime }_1u_1\sim \{u_{j}: 2 \le j \le i \}\sim v_iw_i \sim \{w_{i-j}v^{\prime }_{i-j}v_{i-j} : 1\le j\le i-2 \}\sim \\&\quad w_1v_1\sim \{u_{n-j}v^{\prime }_{n-j}w_{n-j}v_{n-j}:0\le j\le n-i-1 \}\sim v^{\prime }_i=y \end{aligned}$$

Case 10: $x=v^{\prime }_1$ and $y=w_i$

Subcase 10.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v^{\prime }_1v_1u_1 \sim \{u_{j}v_{j}w_{j} v^{\prime }_{j}: 2\le j\le n \} \sim w_1=y \end{aligned}$$

Subcase 10.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=v^{\prime }_1v_1u_1 \sim \{u_{j}v_{j} v^{\prime }_{j}: 2\le j\le i-1 \} \sim \{w_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}:0\le j\le n-i-1 \}\sim v^{\prime }_iu_iv_i w_i=y \end{aligned}$$

Case 11: $x=v^{\prime }_1$ and $y=v_i$

Subcase 11.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v^{\prime }_1u_1 \sim \{u_jv_jw_{j}v^{\prime }_j: 2\le j\le n \} \sim w_1v_1=y \end{aligned}$$

Subcase 11.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=v^{\prime }_1v_1u_1 \sim \{u_{j}v_{j} v^{\prime }_{j}: 2\le j\le i-1 \} \sim \{w_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}: 0\le j\le n-i-1 \}\sim w_iv^{\prime }_iu_iv_i=y \end{aligned}$$

Case 12: $x=v^{\prime }_1$ and $y=u_i$

Subcase 12.1: $i=1$

$$\begin{aligned} P_H(x,y):x=v^{\prime }_1v_1w_1 \sim \{w_jv_ju_{j}v^{\prime }_j: 2\le j\le n \} \sim u_1=y \end{aligned}$$

Subcase 12.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=v^{\prime }_1v_1w_1 \sim \{w_{j}v_{j} v^{\prime }_{j}: 2\le j\le i-1 \} \sim \{u_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{u_{n-j}v^{\prime }_{n-j}w_{n-j}v_{n-j}: 0\le j\le n-i-1 \}\sim w_iv^{\prime }_iv_iu_i=y \end{aligned}$$

Case 13: $x=u_1$ and $y=u_i$, $i\ne 1$

Subcase 13.1: $i=2$

$$\begin{aligned} P_H(x,y):x=u_1v_1w_1 v^{\prime }_1\sim \{v_{j}w_{j} v^{\prime }_{j}: 2\le j\le n \}\sim \{u_{n-j}: 0 \le j \le n-3 \} \sim u_2=y \end{aligned}$$

Subcase 13.2: $i\ge 3$

$$\begin{aligned}&P_H(x,y):x=u_1v_1v^{\prime }_1\sim \{v_{j}u_{j}v^{\prime }_j: 2 \le j \le i-1 \}\sim \{w_{i-j} : 1\le j\le i-1 \}\sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}:0\le j\le n-i-1 \}\sim w_i v^{\prime }_i v_iu_i=y \end{aligned}$$

Case 14: $x=u_1$ and $y=w_i$

Subcase 14.1: $i=1$

$$\begin{aligned} P_H(x,y):x=u_1v_1v^{\prime }_1\sim \{v_ju_{j}v^{\prime }_jw_j: 2\le j\le n \} \sim w_1=y \end{aligned}$$

Subcase 14.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=u_1v_1v^{\prime }_1 \sim \{v_{j}u_{j} v^{\prime }_{j}: 2\le j\le i-1 \} \sim \{w_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}: 0\le j\le n-i-1 \}\sim v^{\prime }_iu_iv_iw_i=y \end{aligned}$$

Case 15: $x=u_1$ and $y=v_i$

Subcase 15.1: $i=1$

$$\begin{aligned} P_H(x,y):x=u_1v^{\prime }_1w_1\sim \{w_jv_{j}u_jv^{\prime }_j: 2\le j\le n \} \sim v_1=y \end{aligned}$$

Subcase 15.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=u_1v_1v^{\prime }_1 \sim \{v_{j}u_{j} v^{\prime }_{j}: 2\le j\le i-1 \} \sim \{w_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}: 0\le j\le n-i-1 \}\sim w_i v^{\prime }_iu_iv_i=y \end{aligned}$$

Case 16: $x=u_1$ and $y=v^{\prime }_i$

Subcase 16.1: $i=1$

$$\begin{aligned} P_H(x,y):x=u_1\sim \{u_jv_{j}w_jv^{\prime }_j: 2\le j\le n \} \sim v_1w_1v^{\prime }_1=y \end{aligned}$$

Subcase 16.2: $i\ge 2$

$$\begin{aligned}&P_H(x,y):x=u_1v_1v^{\prime }_1 \sim \{v_{j}u_{j} v^{\prime }_{j}: 2\le j\le i-1 \}\\&\quad \sim u_iv_iw_i \sim \{w_{i-j}: 1 \le j \le i-1 \}\sim \\&\quad \{v^{\prime }_{n-j}u_{n-j}v_{n-j}w_{n-j}: 0\le j\le n-i-1 \}\sim v^{\prime }_i=y \end{aligned}$$

$\square$