vertex cover

abbr: vc

equivalent to: distance to edgeless, maximum matching

Definition: The minimum number of vertices that have to be removed to get an independent set.

Relations

Other		Relation from	Relation to
acyclic chromatic number	red■	exclusion	upper bound
arboricity	red■	exclusion	upper bound
average degree	red■	exclusion	upper bound
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	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	blue■	exclusion	exclusion
boxicity	red■	exclusion	upper bound
branch width	red■	exclusion	upper bound
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	blue■	exclusion	exclusion
clique-tree-width	red■	exclusion	upper bound
clique-width	red■	exclusion	upper bound
cluster	blue■	unbounded	exclusion
co-cluster	blue■	unbounded	exclusion
cograph	blue■	unbounded	exclusion
complete	blue■	unbounded	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	red■	exclusion	upper bound
degeneracy	red■	exclusion	upper bound
degree treewidth	blue■	exclusion	exclusion
diameter	red■	exclusion	upper bound
diameter+max degree	blue■	exclusion	exclusion
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	blue■	unbounded	exclusion
distance to bipartite	red■	exclusion	upper bound
distance to block	red■	exclusion	upper bound
distance to bounded components	red■	exclusion	upper bound
distance to chordal	red■	exclusion	upper bound
distance to cluster	red■	exclusion	upper bound
distance to co-cluster	red■	exclusion	upper bound
distance to cograph	red■	exclusion	upper bound
distance to complete	blue■	exclusion	exclusion
distance to disconnected	red■	exclusion	upper bound
distance to edgeless	yellow■	equal	equal
distance to forest	red■	exclusion	upper bound
distance to interval	red■	exclusion	upper bound
distance to linear forest	red■	exclusion	upper bound
distance to maximum degree	red■	exclusion	upper bound
distance to outerplanar	red■	exclusion	upper bound
distance to perfect	red■	exclusion	upper bound
distance to planar	red■	exclusion	upper bound
distance to stars	red■	exclusion	upper bound
domatic number	red■	exclusion	upper bound
domination number	blue■	exclusion	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	red■	exclusion	upper bound
edgeless	green■	upper bound	exclusion
feedback edge set	blue■	exclusion	exclusion
feedback vertex set	red■	exclusion	upper bound
forest	blue■	unbounded	exclusion
genus	blue■	exclusion	exclusion
girth	red■	exclusion	upper bound
grid	blue■	unbounded	exclusion
h-index	red■	exclusion	upper bound
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	red■	exclusion	upper bound
maximum degree	blue■	exclusion	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	red■	exclusion	upper bound
maximum leaf number	blue■	exclusion	exclusion
maximum matching	yellow■	upper bound	tight bounds
maximum matching on bipartite graphs	green■	tight bounds	exclusion
mim-width	red■	exclusion	upper bound
minimum degree	red■	exclusion	upper bound
mm-width	red■	exclusion	upper bound
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	red■	exclusion	upper bound
outerplanar	blue■	unbounded	exclusion
path	blue■	unbounded	exclusion
pathwidth	red■	exclusion	upper bound
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	green■	upper bound	exclusion
stars	blue■	unbounded	exclusion
topological bandwidth	blue■	exclusion	exclusion
tree	blue■	unbounded	exclusion
tree-independence number	red■	exclusion	upper bound
treedepth	red■	exclusion	upper bound
treelength	red■	exclusion	upper bound
treewidth	red■	exclusion	upper bound
twin-cover number	red■	exclusion	upper bound
twin-width	red■	exclusion	upper bound
vertex connectivity	red■	exclusion	upper bound
vertex cover	yellow■	equal	equal
vertex integrity	red■	exclusion	upper bound

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 unbounded vertex cover – Note that a clique of size $n$ has … a vertex cover number of $n-1$
- page 23 : vertex cover upper bounds neighborhood diversity by an exponential function – Proposition 4.3. Vertex Cover Number strictly upper bounds Neighborhood Diversity.
- page 23 : bounded neighborhood diversity does not imply bounded vertex cover – Proposition 4.3. Vertex Cover Number strictly upper bounds Neighborhood Diversity.
- 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 34 : bounded c-closure does not imply bounded vertex cover – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
- page 34 : bounded vertex cover does not imply bounded c-closure – Proposition 5.3. $c$-Closure is incomparable to Vertex Cover Number.
2012 Twin-Cover: Beyond Vertex Cover in Parameterized Algorithmics by Ganian
- 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.
assumed
- vertex cover upper bounds twin-cover number by a linear function – By definition
- size upper bounds vertex cover by a linear function – By definition
- vertex cover is equivalent to vertex cover – assumed
unknown source
- maximum matching on bipartite graphs upper and lower bounds vertex cover by a linear function – Kőnig’s theorem
- vertex cover upper and lower bounds maximum matching by a linear function – Every edge of the matching needs to be covered by at least one vertex. Path shows that this relation is not better than linear.
- maximum matching upper bounds vertex cover by a linear function – A set of all vertices taking part in a maximum matching creates a vertex cover, hence $vc(G) \le 2 \cdot mm(G)$.
- vertex cover upper bounds neighborhood diversity by an exponential function
- vertex cover is equivalent to distance to edgeless
- distance to edgeless is equivalent to vertex cover
- edgeless upper bounds vertex cover by a constant
- star upper bounds vertex cover by a constant – trivially
Comparing Graph Parameters by Schröder
- page 15 : bounded vertex cover does not imply bounded domination number – Proposition 3.5
- page 24 : bounded vertex cover does not imply bounded genus – Proposition 3.18
- page 24 : bounded vertex cover does not imply bounded maximum degree – Proposition 3.19
- page 24 : bounded vertex cover does not imply bounded bisection bandwidth – Proposition 3.20