$x=a$

$C_{1},\ldots,C_{k}\cup(C\setminus\{v\})$

$z^{v}$

$\{2,\dots,s-1\}$

$S\cap(A\cup C)$

$H:=G[A\cup B\cup S]$

$z\in V(C)$

$N_{G}(p_{s})\cap V(H)=\{b_{2}\}$

$H\setminus C$

$k\geq\chi(\overline{G})\geq 2$

$H_{1},H_{2}\in\mathcal{M}_{\mathcal{C}}$

$\{p_{0},\dots,p_{s-1}\}$

$(x,c)$

${\cal G}_{1}$

$q_{t}=y$

$q_{2},\dots,q_{t}\in B)$

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

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

$X_{v}$

$\mathcal{M}_{\mathcal{G}_{2}}=\{\overline{K_{2}\cup C_{2k+1}}\mid k\in\mathbb{% N}\}$

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

$\mathcal{O}(n^{\omega}\log n)$

$|V(H)|\geq 2$

$V(C)$

$q_{i},\ldots,q_{t},b_{1},b_{2},a_{2},q_{i}$

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

$G[N_{G}(v)]$

$G_{A}\setminus S^{\prime}$

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

$\mathbb{Z}_{k}$

$\{X_{v}\}_{v\in V(H)}$

$2K_{1}\vee K_{2}$

$G[\{a_{1},a_{2},b_{1},b_{2}\}\cup V(Q)]$

$v^{xy}\in S$

$S\cap A$

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

$X_{w}\subseteq V(D)$

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

$N_{G}[u]=V(G)$

$\mathcal{O}(n^{2.373}(n+m))$

$F\in{\cal F}$

$r_{i}\in C$

$G\setminus x$

$G\setminus N_{G}[v]$

$S^{*}:=(S\setminus\{v^{xy}\})\cup\{x,y\}$

$U:=\{v\in V(H)\mid X_{v}\cap V(C)\neq\emptyset\}$

$N_{A}\cap N_{B}=\emptyset$

$G\setminus(A\cup B)$

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

$S\subseteq S^{*}$

$K_{\ell+1}\in\mathcal{G}_{\mathcal{C}}$

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

$V(H)=A\cup B$

$K_{\ell}$

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

$X_{w}$

$H\setminus a$

$H[A]$

$G^{\prime}[S]$

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

$S^{*}\subseteq S\cup(V(G)\setminus V(H))$

$G^{\prime}:=G/xy$

$p_{i},\dots,p_{s},b_{1},b_{3},a_{3},a_{2},q_{t},\dots,q_{0},p_{i}$

$q_{2},\dots,q_{t}\in Y$

$a_{1},q_{i}$

$\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}}\}$

$p_{1},\dots,p_{i-1}\in A$

$v^{xy}$

$\mathcal{C}\subseteq\mathcal{G}_{\mathcal{C}}$

$S^{\prime}\setminus\{x,y\}$

$p_{1}=x$

$N_{G}(q_{t})\cap V(H)=\{a_{2}\}$

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

$\overline{\overline{G}}=G$

$H[S]\in\mathcal{C}$

$S^{\prime}=S$

$\bigcup_{v\in V(H)}X_{v}\not\subseteq V(H_{2})$

$c\in C\setminus S$

$N_{G}(p_{s})\cap V(H)=\{b_{1}\}$

$\{2K_{1}\vee H\mid H\in\mathcal{M}_{\mathcal{C}_{0}}\}=\{2K_{1}\vee K_{1}\}=\{% P_{3}\}$

$K_{\ell+1}$

$H\in\mathcal{H}$

$A=C\cup\{v\}$

$U=\{v\in V(H)\mid X_{v}\subseteq V(C)\}$

$S_{A}:=S\setminus B$

$2K_{1}\vee 3K_{1}$

$q_{j}\in C$

$p_{i-1},p_{j+1}\in C$

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

$Y\subseteq V(G)\setminus\{x\}$

$\{a,b\}=\{x,y\}$

$\{X_{v}\}_{v\in\{a,b\}\cup V(H)}$

$|V(H)|=2k$

$(x,b)$

$C\cap S$

$\mathcal{O}(|V(G)|^{2})$

$G[a_{j},a_{k},b_{j},b_{k},q_{i},q_{i+1},\dots,q_{t}]$

$2K_{1}$

$a_{i},a_{j}$

$p_{i}\in B$