$\mathcal{E}_{L}^{(0)}$

$\Delta({\text{UNMARK\_LEFT}})=L_{i}\times\bigg{(}-(S_{LM}-L_{i})-S_{LU}-S_{RM}% +S_{RU}\bigg{)}=L_{i}\times\bigg{(}L_{i}-S_{L}+S_{R}\bigg{)}<0$

$|\overline{V_{m}}|=n-2|E_{m}|=n-2k$

$S_{LU}$

$\mathcal{E}_{L}^{(0)}=n_{L}\times\big{(}\sum_{j=1}^{k}(k+1-j)\;w_{j}\big{)}$

$k\leq\frac{n}{2}$

$e_{i}$

$\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})}% )+w^{*}_{k}$

$S_{RU}$

${n_{L}}{n_{R}}(i(j-1)-ij)={n_{L}}{n_{R}}\times(-i)$

$w(e^{*})$

$S_{R}^{\prime}=R_{2}\geq S_{L}^{\prime}=L_{2}+L_{3}$

$\mathcal{W}(E^{\prime})=\sum_{e\in E^{\prime}\cap\overline{E_{m}}}w(e),\;\;% \mathcal{W}^{*}(E^{\prime})=\sum_{e\in E^{\prime}\cap{E_{m}}}w(e),\;\;\mathcal% {W}^{\prime}(E^{\prime})=\sum_{e_{i}\in\overline{E_{m}}\cap E^{\prime}}% \epsilon_{i}$

$|\Delta E|=n_{L}\times k^{\prime}\times B=(n-(k+k^{\prime}))\times k^{\prime}\times B$

$S_{R}=\sum_{i=1}^{{\mathcal{R}}}R_{i}$

$\mathcal{E}$

$|w_{1}+w^{*}+w_{3}-w_{1}-w^{*}-w_{3}|=0$

$P_{n},n\geq 3$

$\displaystyle>n_{L}\times\big{(}\sum_{j=1}^{k}(k+1-j)w_{j}\big{)}\xrightarrow{% \text{See the proof of Lemma \ref{induction1}}}=\mathcal{E}^{(0)}_{L}>\mathcal% {E}^{(\frac{k}{2})}_{L}$

$e^{*}_{k}=(u_{j},u_{j+1})=e^{*}_{3}=(u_{5},u_{6})$

$n_{R}=0$

$\overline{V_{m}}=\{v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3},v^{\prime}_{4}% ,v^{\prime}_{5},v^{\prime}_{6}\}$

${n_{L}}{n_{R}}\times(({\mathcal{L}}-i)({\mathcal{R}}-j+1)-({\mathcal{L}}-i)({% \mathcal{R}}-j))={n_{L}}{n_{R}}\times({\mathcal{L}}{\mathcal{R}}-{\mathcal{L}}% j+{\mathcal{L}}-i{\mathcal{R}}+ij-i-{\mathcal{L}}{\mathcal{R}}+{\mathcal{L}}j+% i{\mathcal{R}}-ij)={n_{L}}{n_{R}}\times({\mathcal{L}}-i)$

$w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\;\forall e_{i}\in E,\;\epsilon_{i}\in% \mathbb{R}$

$0

$e_{i}\in E_{R}$

$\displaystyle|\Delta E|\geq$

$\epsilon\leq c_{i}$

$\Delta_{1}({\text{MARK\_LEFT}})=c_{1}\times(X)$

$A,B,C,D,x,y$

$(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}-i)({n^{2}_{L}}% \times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{\mathcal{R}}))$

$|w_{1}+w_{2}+w_{3}+w^{*}-w_{1}-w_{2}-w_{3}|=w^{*}$

$v_{3}$

$w^{\prime}(e_{i})=w(e_{i})$

$v_{n_{1}+2}$

$e^{*}=(u,v)$

${e^{*}}$

$e_{i}\in E_{L}$

$S_{RU}=S_{R}$

$n_{L}\times\big{(}(i+1)\;w_{i+1}\big{)}$

$c_{i}=\epsilon_{1}=0.4$

$+L_{i}\times S_{RM}$

$V_{m}=\{u_{1},v_{1}\}$

$R_{i}\times R_{j}\times 2w^{*}$

$\mathcal{E}^{(x

$1\leq j\leq{\mathcal{R}}$

$\mathcal{E}(M^{\prime\prime})=\underbrace{\mathcal{E}(M^{\prime})+c_{1}\times X% }_{=\mathcal{E}(M)}+(c_{2}-c_{1})\times X$

$X=L_{1}\times\bigg{(}S_{LM}+(S_{LU}-L_{1})+S_{RM}-S_{RU}\bigg{)}=L_{1}\times% \bigg{(}S_{LM}+(S_{LU}-L_{1})-S_{R}\bigg{)}$

$|x-A|+|x-A-B|$

$\epsilon_{i}$

${e^{*}}=(u,v)$

$c_{1}+c_{2}\leq 1$

$|a|=-a$

$v^{\prime}_{3}$

$u_{1},u_{2}\in\overline{V_{m}}$

$v_{j}\in V_{R},v_{j}\neq v_{n_{1}+2}$

$u_{j}$

$A+B+C$

$\pi_{v,u}$

$\pi^{\prime\prime}_{v,u}$

$\Delta({\text{UNMARK\_RIGHT}})\leq 0\text{ if }{n_{L}}({\mathcal{L}}-2i)\leq{n% _{R}}({\mathcal{R}}-1)\xrightarrow[]{\text{Rearranging the terms}}i\geq\frac{{% \mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$

$\epsilon$

$-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|$

$\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{\prime}(E^{(v_{i},u_% {j})})$

$S_{L}^{\prime}=S_{L}-L_{1}$

$w(e^{*}_{i})$

$e_{0}$

$y=w_{\frac{k}{2}+1}+w_{\frac{k}{2}+2}+\dots+w_{k+1}$

$\pi^{\prime}_{v,u}$

$\mathcal{E}_{LR}=E_{L}=0$

$x-A+x-A-B

$\epsilon_{2}$

$j\xleftarrow{}j-1$

$\Delta({\text{MARK\_RIGHT}})=R_{i}\times\bigg{(}S_{RM}+(S_{RU}-R_{i})+S_{LM}-S% _{LU}\bigg{)}$

$H=G-\{e_{3}=(v^{\prime}_{3},u_{1}),e^{*}=(u_{1},v_{1}),e_{4}=(v_{1},v^{\prime}% _{4})\}$

$R_{j}$

$S_{RU}

$M_{R}^{*}$

$\Delta_{1}({\text{UNMARK\_LEFT}})=L_{i}\times\epsilon\times w^{*}\times(-(S_{L% }-L_{i})+S_{R})=L_{i}\times\epsilon\times w^{*}\times(-S_{L}+L_{i}+S_{R})$

$P_{n}$

$\displaystyle\mathcal{E}_{L}^{(i)}+n_{L}\times\big{(}(i+1)\;w_{i+1}-(k-i)\;w_{% i+1}\big{)}$

$T$

$e^{*}_{k}\in E_{m}$

$G_{2}$

$S_{RU}>S_{LU}$

$\Delta({\text{UNMARK\_RIGHT}})={n^{2}_{R}}(-2(j-1)+2j-{\mathcal{R}}-1)+{n_{L}}% {n_{R}}({\mathcal{L}}-i-i)={n^{2}_{R}}(1-{\mathcal{R}})+{n_{L}}{n_{R}}({% \mathcal{L}}-2i)$

$w_{1}+w_{2}+w_{3}+w^{*}$

$\left|\sum_{k=i}^{j-1}(w_{k}+\epsilon_{k})-\sum_{k=i}^{j-1}w_{k}\right|=\left|% \sum_{k=i}^{j-1}\epsilon_{k}\right|$

$v_{i}=v_{1}$

$n_{L}=\left|\{v|v\in G_{1}\}\right|,n_{R}=\left|\{v|v\in G_{2}\}\right|$

$\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})$

$\mathcal{E}_{L}^{(i+1)}-\mathcal{E}_{L}^{(i)}<0$

$\displaystyle(n_{L}+n_{R})B=(n-2)B=|\overline{V_{m}}|B$

$M_{L}$

$M_{R}^{*}\xleftarrow[]{}M_{R}^{*}\cup\{e_{i}\}$

$S_{LM}$

$e=e_{3}$

$x

$d_{G}(u,v)$

$c_{j}=|x-w_{0}-w_{1}-\dots-w_{j}|$