edge clique cover number

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	cyan■	unknown to HOPS	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	cyan■	unknown to HOPS	exclusion
co-cluster	cyan■	unknown to HOPS	exclusion
cograph	cyan■	unknown to HOPS	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	cyan■	unknown to HOPS	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	blue■	exclusion	exclusion
edge clique cover number	yellow■	equal	equal
edge connectivity	blue■	exclusion	exclusion
edgeless	cyan■	unknown to HOPS	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	red■	exclusion	upper bound
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	blue■	exclusion	exclusion
maximum matching on bipartite graphs	cyan■	unknown to HOPS	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	red■	exclusion	upper bound
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	cyan■	unknown to HOPS	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	blue■	exclusion	exclusion
twin-width	red■	exclusion	upper bound
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 14 : edge clique cover number – The edge clique cover number $eccn(G)$ of a graph $G$ is the minimum number of complete subgraphs required such that each edge is contained in at least one of them.
- page 22 : edge clique cover number upper bounds neighborhood diversity by an exponential function – Theorem 4.1. Edge Clique Cover Number strictly upper bounds Neighborhood Diversity.
- page 22 : bounded neighborhood diversity does not imply bounded edge clique cover number – Theorem 4.1. Edge Clique Cover Number strictly upper bounds Neighborhood Diversity.
- page 23 : distance to complete upper bounds edge clique cover number by a polynomial function – Proposition 4.2. Disatnce to Clique strictly upper bounds Edge Clique Cover Number.
- page 23 : bounded edge clique cover number does not imply bounded distance to complete – Proposition 4.2. Disatnce to Clique strictly upper bounds Edge Clique Cover Number.
- page 29 : bounded edge clique cover number does not imply bounded vertex cover – Proposition 4.13. Edge Clique Cover Number is incomparable to Vertex Cover Number.
- page 29 : bounded vertex cover does not imply bounded edge clique cover number – Proposition 4.13. Edge Clique Cover Number is incomparable to Vertex Cover Number.
- 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.
- page 29 : bounded edge clique cover number does not imply bounded distance to perfect – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
- page 29 : bounded distance to perfect does not imply bounded edge clique cover number – Proposition 4.15. Edge Clique Cover Number is incomparable to Distance to Perfect.
assumed
- edge clique cover number is equivalent to edge clique cover number – assumed
unknown source
- edge clique cover number upper bounds neighborhood diversity by an exponential function – Label vertices by the cliques they are contained in, each label is its own group in the neighborhood diversity, connect accordingly.
- distance to complete upper bounds edge clique cover number by a polynomial function – Cover the remaining clique, cover each modulator vertex and its neighborhood outside of it with another clique, cover each edge within the modulator by its own edge.