twin-cover number
equivalent to: twin-cover number
Relations
Other | Relation from | Relation to |
---|---|---|
acyclic chromatic number | exclusion | exclusion |
arboricity | exclusion | exclusion |
average degree | exclusion | exclusion |
average distance | exclusion | upper bound |
bandwidth | exclusion | exclusion |
bipartite | unbounded | exclusion |
bipartite number | exclusion | upper bound |
bisection bandwidth | exclusion | exclusion |
block | unbounded | unknown to HOPS |
book thickness | exclusion | exclusion |
boolean width | exclusion | upper bound |
bounded components | exclusion | exclusion |
boxicity | exclusion | upper bound |
branch width | exclusion | exclusion |
c-closure | exclusion | exclusion |
carving-width | exclusion | exclusion |
chordal | unbounded | unknown to HOPS |
chordality | exclusion | upper bound |
chromatic number | exclusion | exclusion |
clique cover number | exclusion | exclusion |
clique-tree-width | exclusion | upper bound |
clique-width | exclusion | upper bound |
cluster | constant | unknown to HOPS |
co-cluster | unbounded | exclusion |
cograph | unbounded | unknown to HOPS |
complete | constant | exclusion |
connected | unbounded | unknown to HOPS |
cutwidth | exclusion | exclusion |
cycle | unbounded | exclusion |
cycles | unbounded | exclusion |
d-path-free | exclusion | exclusion |
degeneracy | exclusion | exclusion |
degree treewidth | exclusion | exclusion |
diameter | exclusion | upper bound |
diameter+max degree | exclusion | exclusion |
disjoint cycles | unbounded | exclusion |
distance to bipartite | exclusion | exclusion |
distance to block | exclusion | upper bound |
distance to bounded components | exclusion | exclusion |
distance to chordal | exclusion | upper bound |
distance to cluster | exclusion | upper bound |
distance to co-cluster | exclusion | exclusion |
distance to cograph | exclusion | unknown to HOPS |
distance to complete | exclusion | exclusion |
distance to edgeless | upper bound | exclusion |
distance to forest | exclusion | exclusion |
distance to interval | exclusion | upper bound |
distance to linear forest | exclusion | exclusion |
distance to maximum degree | exclusion | exclusion |
distance to outerplanar | exclusion | exclusion |
distance to perfect | exclusion | upper bound |
distance to planar | exclusion | unknown to HOPS |
distance to stars | unknown to HOPS | exclusion |
domatic number | exclusion | exclusion |
domination number | exclusion | exclusion |
edge clique cover number | exclusion | exclusion |
edge connectivity | exclusion | exclusion |
edgeless | constant | exclusion |
feedback edge set | exclusion | exclusion |
feedback vertex set | exclusion | exclusion |
forest | unbounded | exclusion |
genus | exclusion | exclusion |
girth | exclusion | upper bound |
grid | unbounded | exclusion |
h-index | exclusion | exclusion |
inf-flip-width | exclusion | upper bound |
interval | unbounded | unknown to HOPS |
iterated type partitions | exclusion | unknown to HOPS |
linear clique-width | exclusion | upper bound |
linear forest | unbounded | exclusion |
linear NLC-width | exclusion | upper bound |
linear rank-width | exclusion | upper bound |
maximum clique | exclusion | exclusion |
maximum degree | exclusion | exclusion |
maximum independent set | exclusion | exclusion |
maximum induced matching | exclusion | unknown to HOPS |
maximum leaf number | exclusion | exclusion |
maximum matching | unknown to HOPS | exclusion |
maximum matching on bipartite graphs | upper bound | exclusion |
mim-width | exclusion | upper bound |
minimum degree | exclusion | exclusion |
mm-width | exclusion | exclusion |
modular-width | exclusion | upper bound |
module-width | exclusion | upper bound |
neighborhood diversity | exclusion | exclusion |
NLC-width | exclusion | upper bound |
NLCT-width | exclusion | upper bound |
odd cycle transversal | exclusion | exclusion |
outerplanar | unbounded | exclusion |
path | unbounded | exclusion |
pathwidth | exclusion | exclusion |
pathwidth+maxdegree | exclusion | exclusion |
perfect | unbounded | unknown to HOPS |
planar | unbounded | exclusion |
radius-r flip-width | exclusion | upper bound |
rank-width | exclusion | upper bound |
shrub-depth | exclusion | upper bound |
sim-width | exclusion | upper bound |
star | constant | exclusion |
stars | unknown to HOPS | exclusion |
topological bandwidth | exclusion | exclusion |
tree | unbounded | exclusion |
tree-independence number | exclusion | unknown to HOPS |
treedepth | exclusion | exclusion |
treelength | exclusion | upper bound |
treewidth | exclusion | exclusion |
twin-width | exclusion | upper bound |
vertex connectivity | unknown to HOPS | unknown to HOPS |
vertex cover | upper bound | exclusion |
vertex integrity | 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 : 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.
- 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 : 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 $k+1$.
- unknown source
- bounded twin-cover number does not imply bounded neighborhood diversity
- twin-cover number – Distance to cluster where all vertices of each clique are siblings.
- graph class cluster has constant twin-cover number
- 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