domination number

Definition: Is the cardinality of a minimum vertex set such that its closed neighborhood contains all vertices of the graph.

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	blue■	exclusion	exclusion
bounded components	blue■	exclusion	exclusion
boxicity	blue■	exclusion	exclusion
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chordal	blue■	unbounded	exclusion
chordality	blue■	exclusion	exclusion
chromatic number	blue■	exclusion	exclusion
clique cover number	green■	upper bound	exclusion
clique-tree-width	blue■	exclusion	exclusion
clique-width	blue■	exclusion	exclusion
cluster	blue■	unbounded	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	cyan■	unknown to HOPS	exclusion
disjoint cycles	blue■	unbounded	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	blue■	exclusion	exclusion
distance to chordal	blue■	exclusion	exclusion
distance to cluster	blue■	exclusion	exclusion
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	blue■	exclusion	exclusion
distance to complete	green■	upper bound	exclusion
distance to disconnected	blue■	exclusion	exclusion
distance to edgeless	blue■	exclusion	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	blue■	exclusion	exclusion
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	blue■	exclusion	exclusion
domatic number	blue■	exclusion	exclusion
domination number	yellow■	equal	equal
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	blue■	exclusion	exclusion
edgeless	blue■	unbounded	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	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	blue■	exclusion	exclusion
linear forest	blue■	unbounded	exclusion
linear NLC-width	blue■	exclusion	exclusion
linear rank-width	blue■	exclusion	exclusion
maximum clique	blue■	exclusion	exclusion
maximum degree	blue■	exclusion	exclusion
maximum independent set	green■	upper bound	exclusion
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	blue■	exclusion	exclusion
maximum matching	blue■	exclusion	exclusion
maximum matching on bipartite graphs	blue■	exclusion	exclusion
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	blue■	exclusion	exclusion
mm-width	blue■	exclusion	exclusion
modular-width	blue■	exclusion	exclusion
module-width	blue■	exclusion	exclusion
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	blue■	exclusion	exclusion
NLCT-width	blue■	exclusion	exclusion
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	magenta■	exclusion	unknown to HOPS
rank-width	blue■	exclusion	exclusion
shrub-depth	blue■	exclusion	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
star	cyan■	unknown to HOPS	exclusion
stars	blue■	unbounded	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	blue■	exclusion	exclusion
twin-width	blue■	exclusion	exclusion
vertex connectivity	blue■	exclusion	exclusion
vertex cover	blue■	exclusion	exclusion
vertex integrity	blue■	exclusion	exclusion

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 29 : bounded edge clique cover number does not imply bounded domination number – Proposition 4.14. Edge Clique Cover Number is incomparable to Domination Number.
- page 29 : bounded domination number does not imply bounded edge clique cover number – Proposition 4.14. Edge Clique Cover Number is incomparable to Domination Number.
assumed
- domination number is equivalent to domination number – assumed
Comparing Graph Parameters by Schröder
- page 15 : bounded vertex cover does not imply bounded domination number – Proposition 3.5
unknown source
- maximum independent set upper bounds domination number by a linear function – Every maximal independent set is also a dominating set because any undominated vertex could be added to the independent set.
- domination number upper bounds diameter by a linear function – An unbounded diameter implies a long path where no vertices that are more than $3$ apart may be dominated by the same dominating vertex, otherwise we could shorten the path. Hence, the number of dominating vertices is also unbounded.
- graph class cluster has unbounded domination number
- graph class edgeless has unbounded domination number
- bounded domination number does not imply bounded maximum independent set