distance to cluster

providers: ISGCI

Definition: Minimum number of vertices removed to make the graph into cluster

Relations

Other		Relation from	Relation to
acyclic chromatic number	blue■	exclusion	exclusion
admissibility	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	blue■	exclusion	exclusion
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
bounded expansion	blue■	exclusion	avoids
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chi-bounded	red■	exclusion	upper bound
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■	exclusion	avoids
contraction complexity	blue■	exclusion	exclusion
cutwidth	blue■	exclusion	exclusion
cycle	cyan■	unknown to HOPS	exclusion
cycles	blue■	unbounded	exclusion
d-admissibility	magenta■	exclusion	unknown to HOPS
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
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	yellow■	equal	equal
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	red■	exclusion	upper bound
distance to complete	green■	upper bound	exclusion
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
domino treewidth	blue■	exclusion	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	blue■	exclusion	exclusion
edge-cut width	blue■	exclusion	exclusion
edge-treewidth	blue■	exclusion	exclusion
edgeless	green■	upper bound	avoids
excluded minor	blue■	exclusion	avoids
excluded planar minor	cyan■	unknown to HOPS	avoids
excluded top-minor	blue■	exclusion	avoids
feedback edge set	blue■	exclusion	exclusion
feedback vertex set	blue■	exclusion	exclusion
flip-width	red■	exclusion	upper bound
forest	blue■	unbounded	exclusion
genus	blue■	exclusion	exclusion
grid	blue■	unbounded	exclusion
h-index	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
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	cyan■	unknown to HOPS	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
merge-width	red■	exclusion	upper bound
mim-width	red■	exclusion	upper bound
minimum degree	blue■	exclusion	exclusion
mm-width	blue■	exclusion	exclusion
modular-width	blue■	exclusion	exclusion
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	magenta■	exclusion	unknown to HOPS
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
nowhere dense	magenta■	exclusion	unknown to HOPS
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	blue■	exclusion	exclusion
path	blue■	unbounded	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	blue■	exclusion	exclusion
perfect	blue■	unbounded	exclusion
planar	blue■	unbounded	exclusion
radius-inf flip-width	red■	exclusion	upper bound
radius-r flip-width	red■	exclusion	upper bound
rank-width	red■	exclusion	upper bound
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	red■	exclusion	upper bound
sim-width	red■	exclusion	upper bound
size	green■	upper bound	exclusion
slim tree-cut width	blue■	exclusion	exclusion
sparse twin-width	blue■	exclusion	exclusion
star	green■	upper bound	exclusion
stars	cyan■	unknown to HOPS	exclusion
strong coloring number	blue■	exclusion	exclusion
strong d-coloring number	magenta■	exclusion	unknown to HOPS
strong inf-coloring number	blue■	exclusion	exclusion
topological bandwidth	blue■	exclusion	exclusion
tree	blue■	unbounded	exclusion
tree-cut width	blue■	exclusion	exclusion
tree-independence number	red■	exclusion	upper bound
tree-partition-width	blue■	exclusion	exclusion
treebandwidth	blue■	exclusion	exclusion
treedepth	blue■	exclusion	exclusion
treelength	red■	exclusion	upper bound
treespan	blue■	exclusion	exclusion
treewidth	blue■	exclusion	exclusion
twin-cover number	green■	upper bound	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	blue■	exclusion	exclusion
weak coloring number	blue■	exclusion	exclusion
weak d-coloring number	magenta■	exclusion	unknown to HOPS
weak inf-coloring number	blue■	exclusion	exclusion
weakly sparse	magenta■	exclusion	unknown to HOPS
weakly sparse and merge width	blue■	exclusion	exclusion

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- 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 : graph classes with bounded distance to cluster are not bounded twin-cover number – We show that twin cover number is not upper bounded by distance to cluster.
- page 28 : graph classes with bounded modular-width are not bounded distance to cluster – Proposition 4.10. Modular-width is incomparable to Distance to Cluster.
- page 28 : graph classes with bounded distance to cluster are not bounded modular-width – Proposition 4.10. Modular-width is incomparable to Distance to Cluster.
Comparing Graph Parameters by Schröder
- page 13 : graph classes with bounded distance to cluster are not bounded distance to co-cluster – Proposition 3.3
unknown source
- graph class path is not constant distance to cluster – trivially
- distance to cluster upper bounds distance to cograph by a linear function
- distance to cluster upper bounds shrub-depth by a constant – J. Pokorný, personal communication: Assume the class of constant dtc we want to show it has constant sd as well. For each clique connect them in a star in the tree model T. Each vertex in the modulator connect to their own vertex in T. Add a root that is in distance 2 to all leaves. Now give each vertex in the modulator a unique colour. Each other vertex that is not in the modulator has as it’s colour the set of neighbours from the modulator. In total there are $2^{dtc} + dtc$ colours that is a constant.
assumed
- cluster upper bounds distance to cluster by a constant – by definition
- twin-cover number upper bounds distance to cluster by a linear function – By definition
- distance to cluster is equivalent to distance to cluster – assumed