size

Definition: Total size of the graph $|V(G)+E(G)|$.

Relations

Other		Relation from	Relation to
acyclic chromatic number	red■	exclusion	upper bound
admissibility	red■	exclusion	upper bound
arboricity	red■	exclusion	upper bound
average degree	red■	exclusion	upper bound
average distance	red■	exclusion	upper bound
bandwidth	red■	exclusion	upper bound
bipartite	blue■	unbounded	exclusion
bipartite number	red■	exclusion	upper bound
bisection bandwidth	red■	exclusion	upper bound
block	blue■	unbounded	exclusion
book thickness	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	red■	exclusion	upper bound
bounded expansion	red■	exclusion	upper bound
boxicity	red■	exclusion	upper bound
branch width	red■	exclusion	upper bound
c-closure	red■	exclusion	upper bound
carving-width	red■	exclusion	upper bound
chi-bounded	red■	exclusion	upper bound
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	red■	exclusion	upper bound
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■	exclusion	avoids
contraction complexity	red■	exclusion	upper bound
cutwidth	red■	exclusion	upper bound
cycle	cyan■	unknown to HOPS	exclusion
cycles	magenta■	unbounded	unknown to HOPS
d-admissibility	red■	exclusion	upper bound
d-path-free	red■	exclusion	upper bound
degeneracy	red■	exclusion	upper bound
degree treewidth	red■	exclusion	upper bound
diameter	red■	exclusion	upper bound
diameter+max degree	red■	exclusion	upper bound
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	red■	exclusion	upper bound
distance to edgeless	red■	exclusion	upper bound
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	red■	exclusion	upper bound
domino treewidth	red■	exclusion	upper bound
edge clique cover number	red■	exclusion	upper bound
edge connectivity	red■	exclusion	upper bound
edge-cut width	red■	exclusion	upper bound
edge-treewidth	red■	exclusion	upper bound
edgeless	blue■	exclusion	avoids
excluded minor	red■	exclusion	upper bound
excluded planar minor	gray■	unknown to HOPS	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	red■	exclusion	upper bound
feedback vertex set	red■	exclusion	upper bound
flip-width	red■	exclusion	upper bound
forest	blue■	unbounded	exclusion
genus	red■	exclusion	upper bound
grid	blue■	unbounded	exclusion
h-index	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	red■	exclusion	upper bound
maximum independent set	red■	exclusion	upper bound
maximum induced matching	red■	exclusion	upper bound
maximum leaf number	orange■	unknown to HOPS	upper bound
maximum matching	red■	exclusion	upper bound
maximum matching on bipartite graphs	blue■	exclusion	exclusion
merge-width	red■	exclusion	upper bound
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
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	red■	exclusion	upper bound
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
nowhere dense	red■	exclusion	upper bound
odd cycle transversal	red■	exclusion	upper bound
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	red■	exclusion	upper bound
path	blue■	unbounded	exclusion
pathwidth	red■	exclusion	upper bound
pathwidth+maxdegree	red■	exclusion	upper bound
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	yellow■	equal	equal
slim tree-cut width	red■	exclusion	upper bound
sparse twin-width	red■	exclusion	upper bound
star	blue■	unbounded	exclusion
stars	blue■	unbounded	exclusion
strong coloring number	red■	exclusion	upper bound
strong d-coloring number	red■	exclusion	upper bound
strong inf-coloring number	red■	exclusion	upper bound
topological bandwidth	red■	exclusion	upper bound
tree	blue■	unbounded	exclusion
tree-cut width	red■	exclusion	upper bound
tree-independence number	red■	exclusion	upper bound
tree-partition-width	red■	exclusion	upper bound
treebandwidth	red■	exclusion	upper bound
treedepth	red■	exclusion	upper bound
treelength	red■	exclusion	upper bound
treespan	red■	exclusion	upper bound
treewidth	red■	exclusion	upper bound
twin-cover number	red■	exclusion	upper bound
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	red■	exclusion	upper bound
vertex integrity	red■	exclusion	upper bound
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	red■	exclusion	upper bound
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

1993 On the chordality of a graph by McKee, Scheinerman
- page 2 : size upper bounds chordality by a constant – Corollary 3. For any graph $G$, $\mathrm{Chord}(G) \le |V(G)|/2$.
assumed
- size upper bounds vertex cover by a linear function – By definition
- size upper bounds maximum leaf number by a linear function – By definition
- size upper bounds distance to complete by a linear function – By definition
- size upper bounds bounded components by a linear function – By definition
- graph classes with bounded size do not have contained in planar – By definition
- graph classes with bounded size do not have contained in perfect – By definition
- graph classes with bounded size are not connected – By definition
- size is equivalent to size – assumed