$G\setminus\{a_{1},b_{1}\}$

$y\in V(F)$

$q_{1}=c$

$v\in V(G)\setminus V(H)$

$G[q_{j},\dots,,q_{t},b_{1},a_{1},a_{2}]$

$N_{G}(q_{t})\cap V(H)=B$

$G[S]=H[S]$

$\omega<2.3728596$

$x\in A$

$G_{i}-v_{i}$

$H\in\mathcal{C}$

$\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{1}}\}=\{2K_{1}\vee 2K_{1}\}=% \{C_{4}\}$

$N_{G}(v)\cap V(H)\subseteq B$

$a_{2},a_{3},b_{1}$

$Q=q_{0},\dots,q_{t}$

$H\cong 2K_{1}$

$\mathcal{G}_{k}$

$V(G)\setminus V(H)$

$E(H_{1})\cup E(H_{2})\cup\{v_{1}v_{2}\mid v_{1}\in V(H_{1}),v_{2}\in V(H_{2})\}$

$G[S^{\prime}]\in\mathcal{C}$

$G_{A}:=G[A\cup C]$

$\bigcup_{v\in V(H)}X_{v}\subseteq V(2K_{1})$

$\mathcal{O}(n^{2}(n+m))$

$K_{2,3}$

$p_{1},\dots,p_{i},q_{j},\dots,q_{t}$

$K_{p,q}$

$\mathcal{F}_{\mathcal{C}}$

$S^{\prime}\subseteq S^{*}$

$G\in\mathcal{C}_{k}$

$\mathcal{O}(n^{4})$

$r_{1},\dots,r_{i},c$

$S=V(G)$

$\mathcal{O}(n^{2k+2})$

$S\subseteq A\cup C$

$S^{\prime}\subseteq S\cap(A\cup C)$

$S\subseteq V(G^{\prime})\setminus\{a,b,v^{xy}\}=V(G)\setminus\{a,b,x,y\}$

$A=\{a_{1},a_{2},a_{3}\}$

$G/e$

$p_{s}=c$

$S\subseteq V(G^{\prime})\setminus\{v^{xy}\}=V(G)\setminus\{x,y\}$

$a_{i},a_{j},a_{k}\in A$

$G[a_{1},a_{2},b_{1},b_{2},p_{0},\dots,p_{s}]$

$x,y\in V(G^{\prime})$

$S\subseteq V(G)\setminus\{a,b\}$

$\sum_{x\in X}w(x)$

$S_{v}=\{v,z^{v}\}$

$\mathcal{O}(n^{2+\epsilon})$

$p_{1},\ldots,p_{k}$

$r_{1}=x$

$K_{2,k+1}\notin\mathcal{G}_{k}$

$p_{1},\dots,p_{i-1},p_{j+1},\dots,p_{s}$

$P^{3}$

$p_{1},\dots,p_{i},q_{j+1},\dots,q_{t}$

$\chi(\overline{G})\leq k$

$r_{i},c\in C$

$G[S^{*}]\in\mathcal{C}$

$F\in{\mathcal{C}}$

$\overline{C_{6}}$

$|V(H)|-1$

$G\setminus S^{\prime}$

$S^{*}\cap V(H)\subseteq S$

$G^{\prime}[S]=G[S]$

$\overline{K_{2}\cup C_{2k+1}}\cong 2K_{1}\vee\overline{C_{2k+1}}$

$S\cap B=\emptyset$

$h_{1},\dots,h_{i}$

$y=b$

$a_{2},q_{j},b_{1}$

$K_{2,k}$

$a,b\in V(G)\setminus S$

$y\in B$

$N_{G}(q_{t})\cap V(H)=\{b_{1}\}$

$\mathcal{M}_{\mathcal{G}_{0}}=\{P_{3}\}$

$\mathcal{O}(n^{3+\epsilon})$

$K_{k-1}$

$\mathcal{O}(n^{2+\epsilon}).$

$p_{i}\in C$

$r_{k}=y$

$K_{2,k+1}$

$V(C_{1})$

$N_{G}[y]\setminus N_{G}[x]$

$p_{s}\in N_{B}$

${\mathcal{C}}$

$N_{G}[x]:=\{x\}\cup N_{G}(x)$

$N_{B}=\emptyset$

$\alpha(G^{*})=\alpha(G)+|E(G)|$

$G[v,a_{i},a_{j},a_{k}]$

$p_{1}\in A$

$G[x_{1},\dots,x_{t}]$

$G^{\prime}\in\mathcal{G}_{\mathcal{C}}$

$K_{\ell}\notin\mathcal{C}$

$V(H)\setminus U$

$G\in\mathcal{C}$

$uw\in E(H)$

$a_{2}\in N_{G}(p_{s})\cap V(H)\subseteq\{a_{2},b_{2}\}$

$G[A\cup B\cup\{q_{0},q_{1}\}]$

$\{a_{i},b_{i}\}$

$|N_{G}(v)\cap V(H)|\geq 2$

$a,b\in V(G^{\prime})\setminus S$

$X=\{x\}$

$E(H_{1})\cup E(H_{2})$