twin-cover number

Definition: In graph $G$, twin-cover number is the minimum number $k$ such that there exists a set $M$ of size $k$ such that $G-M$ is a union of cliques where each pair of vertices from the same clique are true siblings in $G$.

Relations

Other		Relation from	Relation to
acyclic chromatic number	blue■	exclusion	exclusion
arboricity	blue■	exclusion	exclusion
average degree	blue■	exclusion	exclusion
average distance	red■	exclusion	upper bound
bandwidth	blue■	exclusion	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	red■	exclusion	upper bound
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	blue■	exclusion	exclusion
boolean width	red■	exclusion	upper bound
bounded components	blue■	exclusion	exclusion
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	blue■	exclusion	exclusion
clique cover number	blue■	exclusion	exclusion
clique-tree-width	red■	exclusion	upper bound
clique-width	red■	exclusion	upper bound
cluster	green■	upper bound	exclusion
co-cluster	blue■	unbounded	exclusion
cograph	blue■	unbounded	exclusion
complete	green■	upper bound	exclusion
connected	blue■	unbounded	exclusion
contraction complexity	blue■	exclusion	exclusion
cutwidth	blue■	exclusion	exclusion
cycle	blue■	unbounded	exclusion
cycles	blue■	unbounded	exclusion
d-path-free	blue■	exclusion	exclusion
degeneracy	blue■	exclusion	exclusion
degree treewidth	blue■	exclusion	exclusion
diameter	red■	exclusion	upper bound
diameter+max degree	blue■	exclusion	exclusion
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	blue■	unbounded	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	red■	exclusion	upper bound
distance to bounded components	blue■	exclusion	exclusion
distance to chordal	red■	exclusion	upper bound
distance to cluster	red■	exclusion	upper bound
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	red■	exclusion	upper bound
distance to complete	blue■	exclusion	exclusion
distance to disconnected	magenta■	exclusion	unknown to HOPS
distance to edgeless	green■	upper bound	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	red■	exclusion	upper bound
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	red■	exclusion	upper bound
distance to planar	blue■	exclusion	exclusion
distance to stars	cyan■	unknown to HOPS	exclusion
domatic number	blue■	exclusion	exclusion
domination number	blue■	exclusion	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	blue■	exclusion	exclusion
edgeless	green■	upper bound	exclusion
feedback edge set	blue■	exclusion	exclusion
feedback vertex set	blue■	exclusion	exclusion
forest	blue■	unbounded	exclusion
genus	blue■	exclusion	exclusion
girth	red■	exclusion	upper bound
grid	blue■	unbounded	exclusion
h-index	blue■	exclusion	exclusion
inf-flip-width	red■	exclusion	upper bound
interval	blue■	unbounded	exclusion
iterated type partitions	magenta■	exclusion	unknown to HOPS
linear clique-width	red■	exclusion	upper bound
linear forest	blue■	unbounded	exclusion
linear NLC-width	red■	exclusion	upper bound
linear rank-width	red■	exclusion	upper bound
maximum clique	blue■	exclusion	exclusion
maximum degree	blue■	exclusion	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	blue■	exclusion	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
mim-width	red■	exclusion	upper bound
minimum degree	blue■	exclusion	exclusion
mm-width	blue■	exclusion	exclusion
modular-width	red■	exclusion	upper bound
module-width	red■	exclusion	upper bound
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	blue■	unbounded	exclusion
path	blue■	unbounded	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	blue■	exclusion	exclusion
perfect	blue■	unbounded	exclusion
planar	blue■	unbounded	exclusion
radius-r flip-width	red■	exclusion	upper bound
rank-width	red■	exclusion	upper bound
shrub-depth	red■	exclusion	upper bound
sim-width	red■	exclusion	upper bound
size	green■	upper bound	exclusion
star	green■	upper bound	exclusion
stars	cyan■	unknown to HOPS	exclusion
topological bandwidth	blue■	exclusion	exclusion
tree	blue■	unbounded	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
treedepth	blue■	exclusion	exclusion
treelength	red■	exclusion	upper bound
treewidth	blue■	exclusion	exclusion
twin-cover number	yellow■	equal	equal
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	blue■	exclusion	exclusion

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 18 : vertex cover upper bounds twin-cover number by a linear function – By definition
- page 18 : complete upper bounds twin-cover number by a constant – 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.
2015 Improving Vertex Cover as a Graph Parameter by Ganian
- 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.
2013 Parameterized Algorithms for Modular-Width by Gajarský, Lampis, Ordyniak
- 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, …
2012 Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics by Ganian
- 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 : complete upper bounds twin-cover number by a constant – 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 constant – 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 constant – 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 constant – The rank-width and linaer rank-width of graph of twin-cover $k$ are at most $k+1$.
unknown source
- bounded twin-cover number does not imply bounded neighborhood diversity
- cluster upper bounds twin-cover number by a constant
assumed
- twin-cover number upper bounds distance to cluster by a linear function – By definition
- vertex cover upper bounds twin-cover number by a linear function – By definition
- twin-cover number is equivalent to twin-cover number – assumed