domatic number
equivalent to: domatic number
Relations
Other ⮁ | Relation from ⮁ | Relation to ⮁ |
---|---|---|
acyclic chromatic number | upper bound | exclusion |
arboricity | upper bound | exclusion |
average degree | upper bound | unknown to HOPS |
average distance | exclusion | exclusion |
bandwidth | upper bound | exclusion |
bipartite | unknown to HOPS | exclusion |
bipartite number | exclusion | unknown to HOPS |
bisection bandwidth | exclusion | exclusion |
block | unbounded | exclusion |
book thickness | upper bound | exclusion |
boolean width | exclusion | exclusion |
bounded components | upper bound | exclusion |
boxicity | exclusion | exclusion |
branch width | upper bound | exclusion |
c-closure | exclusion | exclusion |
carving-width | upper bound | exclusion |
chordal | unbounded | exclusion |
chordality | exclusion | exclusion |
chromatic number | exclusion | exclusion |
clique cover number | exclusion | exclusion |
clique-tree-width | exclusion | exclusion |
clique-width | exclusion | exclusion |
cluster | unbounded | exclusion |
co-cluster | unbounded | exclusion |
cograph | unbounded | exclusion |
complete | unbounded | exclusion |
connected | unbounded | unknown to HOPS |
cutwidth | upper bound | exclusion |
cycle | constant | exclusion |
cycles | constant | exclusion |
d-path-free | upper bound | exclusion |
degeneracy | upper bound | exclusion |
degree treewidth | upper bound | exclusion |
diameter | exclusion | exclusion |
diameter+max degree | upper bound | exclusion |
disjoint cycles | constant | exclusion |
distance to bipartite | exclusion | exclusion |
distance to block | exclusion | exclusion |
distance to bounded components | upper bound | exclusion |
distance to chordal | exclusion | exclusion |
distance to cluster | exclusion | exclusion |
distance to co-cluster | exclusion | exclusion |
distance to cograph | exclusion | exclusion |
distance to complete | exclusion | exclusion |
distance to edgeless | upper bound | exclusion |
distance to forest | upper bound | exclusion |
distance to interval | exclusion | exclusion |
distance to linear forest | upper bound | exclusion |
distance to maximum degree | upper bound | exclusion |
distance to outerplanar | upper bound | exclusion |
distance to perfect | exclusion | exclusion |
distance to planar | unknown to HOPS | exclusion |
distance to stars | upper bound | exclusion |
domination number | exclusion | exclusion |
edge clique cover number | exclusion | exclusion |
edge connectivity | exclusion | unknown to HOPS |
edgeless | constant | exclusion |
feedback edge set | upper bound | exclusion |
feedback vertex set | upper bound | exclusion |
forest | constant | exclusion |
genus | upper bound | exclusion |
girth | exclusion | exclusion |
grid | constant | exclusion |
h-index | upper bound | exclusion |
inf-flip-width | exclusion | exclusion |
interval | unbounded | exclusion |
iterated type partitions | exclusion | exclusion |
linear clique-width | exclusion | exclusion |
linear forest | constant | exclusion |
linear NLC-width | exclusion | exclusion |
linear rank-width | exclusion | exclusion |
maximum clique | exclusion | exclusion |
maximum degree | upper bound | exclusion |
maximum independent set | exclusion | exclusion |
maximum induced matching | exclusion | exclusion |
maximum leaf number | upper bound | exclusion |
maximum matching | unknown to HOPS | exclusion |
maximum matching on bipartite graphs | upper bound | exclusion |
mim-width | exclusion | unknown to HOPS |
minimum degree | upper bound | unknown to HOPS |
mm-width | upper bound | exclusion |
modular-width | exclusion | exclusion |
module-width | exclusion | exclusion |
neighborhood diversity | exclusion | exclusion |
NLC-width | exclusion | exclusion |
NLCT-width | exclusion | exclusion |
odd cycle transversal | exclusion | exclusion |
outerplanar | constant | exclusion |
path | constant | exclusion |
pathwidth | upper bound | exclusion |
pathwidth+maxdegree | upper bound | exclusion |
perfect | unbounded | exclusion |
planar | constant | exclusion |
radius-r flip-width | exclusion | unknown to HOPS |
rank-width | exclusion | exclusion |
shrub-depth | exclusion | exclusion |
sim-width | exclusion | unknown to HOPS |
star | constant | exclusion |
stars | constant | exclusion |
topological bandwidth | upper bound | exclusion |
tree | constant | exclusion |
tree-independence number | unknown to HOPS | unknown to HOPS |
treedepth | upper bound | exclusion |
treelength | exclusion | unknown to HOPS |
treewidth | upper bound | exclusion |
twin-cover number | exclusion | exclusion |
twin-width | exclusion | exclusion |
vertex connectivity | unknown to HOPS | exclusion |
vertex cover | upper bound | exclusion |
vertex integrity | upper bound | exclusion |
Results
- 2022 Expanding the Graph Parameter Hierarchy by Tran
- 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 34 : bounded c-closure does not imply bounded domatic number – Observation 5.7.
-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique. - page 34 : bounded domatic number does not imply bounded c-closure – Observation 5.7.
-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
- Comparing Graph Parameters by Schröder
- page 16 : bounded distance to complete does not imply bounded domatic number – Proposition 3.7
- page 20 : bounded distance to co-cluster does not imply bounded domatic number – Proposition 3.12
- page 20 : bounded distance to bipartite does not imply bounded domatic number – Proposition 3.12
- page 30 : bounded bisection bandwidth does not imply bounded domatic number – Proposition 3.28
- page 31 : bounded domatic number does not imply bounded vertex connectivity – Proposition 3.30
- unknown source
- minimum degree upper bounds domatic number by a linear function – The vertex of minimum degree needs to be dominated in each of the sets. As the sets cannot overlap there can be at most
of them. - graph class complete has unbounded domatic number – Parameter is unbounded for the graph class of cliques.
- minimum degree upper bounds domatic number by a linear function – The vertex of minimum degree needs to be dominated in each of the sets. As the sets cannot overlap there can be at most
- https://mathworld.wolfram.com/DomaticNumber.html
- domatic number – The maximum number of disjoint dominating sets in a domatic partition of a graph
is called its domatic number .
- domatic number – The maximum number of disjoint dominating sets in a domatic partition of a graph