page 5 : twin-cover number – Definition 3 A set of vertices $X \subseteq V(G)$ is a twin-cover of $G$ if for every edge $e = ab \in E(G)$ either 1. $a \in X$ or $b \in X$, or 2. $a$ and $b$ are true twins. We then say that $G$ has twin-cover $k$ if the size of a minimum twin-cover of $G$ is $k$.
page 20 : twin-cover number upper bounds shrub-depth by a constant – Let $\mathcal H_k$ be the class of graphs of twin-cover $k$. Then $\mathcal H_k \subseteq \mathcal{TM}_{2^k+k}(2)$ and a tree-model of any $G \in \mathcal H_k$ may be constructed in single-exponential FPT time.
page 262 : twin-cover number – Definition 3.1. $X \subseteq V(G)$ is a twin-cover of $G$ if for every edge $e={a,b} \in E(G)$ either 1. $a \in X$ or $b \in X$, or 2. $a$ and $b$ are twins, i.e. all other vertices are either adjacent to both $a$ and $b$ or none. We then say that $G$ has twin-cover number $k$ if $k$ is the minimum possible size of a twin-cover of $G$.
page 262 : twin-cover number – Definition 3.2. $X \subseteq V(G)$ is a twin-cover of $G$ if there exists a subgraph $G’$ of $G$ such that 1. $X \subseteq V(G’)$ and $X$ is a vertex cover of $G’$. 2. $G$ can be obtained by iteratively adding twins to non-cover vertices in $G’$.
page 263 : graph class complete has constant twin-cover number – We note that complete graphs indeed have a twin-cover of zero.
page 263 : bounded twin-cover number does not imply bounded vertex cover – The vertex cover of graphs of bounded twin-cover may be arbitrarily large.
page 263 : bounded twin-cover number does not imply bounded treewidth – There exists graphs with arbitrarily large twin-cover and bounded tree-width and vice-versa.
page 263 : bounded treewidth does not imply bounded twin-cover number – There exists graphs with arbitrarily large twin-cover and bounded tree-width and vice-versa.
page 263 : twin-cover number upper bounds clique-width by a linear function – The clique-width of graphs of twin-cover $k$ is at most $k+2$.
page 263 : twin-cover number upper bounds rank-width by a linear function – The rank-width and linaer rank-width of graph of twin-cover $k$ are at most $k+1$.
page 263 : twin-cover number upper bounds linear rank-width by a linear function – The rank-width and linaer rank-width of graph of twin-cover $k$ are at most $k+1$.
page 6 : twin-cover number upper bounds modular-width by an exponential function – Theorem 3. Let $G$ be a graph. Then $\mathrm{mw}(G) \le \mathrm{nd}(G)$ and $\mathrm{mw}(G) \le 2\mathrm{tc}(G) + \mathrm{tc}(G)$. Furthermore, both inequalities are strict, …
page 6 : bounded modular-width does not imply bounded twin-cover number – Theorem 3. Let $G$ be a graph. Then $\mathrm{mw}(G) \le \mathrm{nd}(G)$ and $\mathrm{mw}(G) \le 2\mathrm{tc}(G) + \mathrm{tc}(G)$. Furthermore, both inequalities are strict, …
page 18 : graph class complete has constant twin-cover number – Note that a clique of size $n$ has a twin cover number of 0 …
page 18 : twin-cover number upper bounds distance to cluster by a linear function – … graph $H$ with a twin cover of size $k$ has a distance to cluster of at most $k$.
page 18 : bounded distance to cluster does not imply bounded twin-cover number – We show that twin cover number is not upper bounded by distance to cluster.
page 18 : bounded twin-cover number does not imply bounded distance to complete – Observation 3.3. Twin Cover Number is incomparable to Distance to Clique.
page 18 : bounded distance to complete does not imply bounded twin-cover number – Observation 3.3. Twin Cover Number is incomparable to Distance to Clique.
page 19 : bounded twin-cover number does not imply bounded maximum clique – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
page 19 : bounded maximum clique does not imply bounded twin-cover number – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
page 19 : bounded twin-cover number does not imply bounded domatic number – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
page 19 : bounded domatic number does not imply bounded twin-cover number – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
page 19 : bounded twin-cover number does not imply bounded edge connectivity – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
page 19 : bounded edge connectivity does not imply bounded twin-cover number – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
page 21 : bounded twin-cover number does not imply bounded distance to co-cluster – Proposition 3.5. Twin Cover Number is incomparable to Distance to Co-Cluster.
page 21 : bounded distance to co-cluster does not imply bounded twin-cover number – Proposition 3.5. Twin Cover Number is incomparable to Distance to Co-Cluster.
page 28 : bounded neighborhood diversity does not imply bounded twin-cover number – Proposition 4.12. Modular-width is incomparable to Distance to Twin Cover Number.
page 28 : bounded twin-cover number does not imply bounded neighborhood diversity – Proposition 4.12. Modular-width is incomparable to Distance to Twin Cover Number.
