planar

providers: ISGCI

Definition: Has an embedding with vertices being points, edges being curves between respective points, which is planar, i.e., no curves cross each other.

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	blue■	exclusion	unbounded
bandwidth	blue■	exclusion	unbounded
bipartite	blue■	exclusion	exclusion
bipartite number	magenta■	exclusion	unknown to HOPS
bisection bandwidth	blue■	exclusion	unbounded
block	blue■	exclusion	exclusion
book thickness	red■	exclusion	upper bound
boolean width	blue■	exclusion	unbounded
bounded components	blue■	exclusion	unbounded
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	unbounded
c-closure	magenta■	exclusion	unknown to HOPS
carving-width	blue■	exclusion	unbounded
chordal	blue■	exclusion	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	blue■	exclusion	unbounded
clique-tree-width	blue■	exclusion	unbounded
clique-width	blue■	exclusion	unbounded
cluster	blue■	exclusion	exclusion
co-cluster	blue■	exclusion	exclusion
cograph	blue■	exclusion	exclusion
complete	blue■	exclusion	exclusion
connected	blue■	exclusion	exclusion
contraction complexity	blue■	exclusion	unbounded
cutwidth	blue■	exclusion	unbounded
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-path-free	blue■	exclusion	unbounded
degeneracy	red■	exclusion	upper bound
degree treewidth	blue■	exclusion	unbounded
diameter	blue■	exclusion	unbounded
diameter+max degree	blue■	exclusion	unbounded
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	green■	upper bound	exclusion
distance to bipartite	blue■	exclusion	unbounded
distance to block	blue■	exclusion	unbounded
distance to bounded components	blue■	exclusion	unbounded
distance to chordal	blue■	exclusion	unbounded
distance to cluster	blue■	exclusion	unbounded
distance to co-cluster	blue■	exclusion	unbounded
distance to cograph	blue■	exclusion	unbounded
distance to complete	blue■	exclusion	unbounded
distance to disconnected	red■	exclusion	upper bound
distance to edgeless	blue■	exclusion	unbounded
distance to forest	blue■	exclusion	unbounded
distance to interval	blue■	exclusion	unbounded
distance to linear forest	blue■	exclusion	unbounded
distance to maximum degree	blue■	exclusion	unbounded
distance to outerplanar	blue■	exclusion	unbounded
distance to perfect	blue■	exclusion	unbounded
distance to planar	red■	exclusion	upper bound
distance to stars	blue■	exclusion	unbounded
domatic number	red■	exclusion	upper bound
domination number	blue■	exclusion	unbounded
edge clique cover number	blue■	exclusion	unbounded
edge connectivity	red■	exclusion	upper bound
edgeless	green■	upper bound	exclusion
feedback edge set	blue■	exclusion	unbounded
feedback vertex set	blue■	exclusion	unbounded
forest	green■	upper bound	exclusion
genus	red■	exclusion	upper bound
girth	blue■	exclusion	unbounded
grid	green■	inclusion	exclusion
h-index	blue■	exclusion	unbounded
inf-flip-width	blue■	exclusion	unbounded
interval	blue■	exclusion	exclusion
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	blue■	exclusion	unbounded
linear forest	green■	upper bound	exclusion
linear NLC-width	blue■	exclusion	unbounded
linear rank-width	blue■	exclusion	unbounded
maximum clique	red■	exclusion	upper bound
maximum degree	blue■	exclusion	unbounded
maximum independent set	blue■	exclusion	unbounded
maximum induced matching	blue■	exclusion	unbounded
maximum leaf number	blue■	exclusion	unbounded
maximum matching	blue■	exclusion	unbounded
maximum matching on bipartite graphs	cyan■	unknown to HOPS	unbounded
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	red■	exclusion	upper bound
mm-width	blue■	exclusion	unbounded
modular-width	blue■	exclusion	unbounded
module-width	blue■	exclusion	unbounded
neighborhood diversity	blue■	exclusion	unbounded
NLC-width	blue■	exclusion	unbounded
NLCT-width	blue■	exclusion	unbounded
odd cycle transversal	blue■	exclusion	unbounded
outerplanar	green■	inclusion	exclusion
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	blue■	exclusion	exclusion
planar	yellow■	equal	equal
radius-r flip-width	red■	exclusion	upper bound
rank-width	blue■	exclusion	unbounded
shrub-depth	blue■	exclusion	unbounded
sim-width	magenta■	exclusion	unknown to HOPS
size	blue■	exclusion	unbounded
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
topological bandwidth	blue■	exclusion	unbounded
tree	green■	upper bound	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
treedepth	blue■	exclusion	unbounded
treelength	magenta■	exclusion	unknown to HOPS
treewidth	blue■	exclusion	unbounded
twin-cover number	blue■	exclusion	unbounded
twin-width	red■	exclusion	upper bound
vertex connectivity	red■	exclusion	upper bound
vertex cover	blue■	exclusion	unbounded
vertex integrity	blue■	exclusion	unbounded

Results

2017 Graph Theory by Diestel
- page 89 : planar – When we draw a graph on a piece of paper, we naturally try to do this as transparently as possible. One obvious way to limit the mess created by all the lines is to avoid intersections. … Graphs drawn in this way are called \emph{plane graphs}; abstract graphs that can be drawn in this way are called \emph{planar}.
unknown source
- planar upper bounds genus by a constant
- graph class planar has unbounded girth
- graph class planar has unbounded maximum degree
- graph class planar has unbounded distance to perfect
assumed
- graph class outerplanar is included in graph class planar
- graph class planar is not included in graph class outerplanar
- graph class grid is included in graph class planar
- graph class planar is not included in graph class grid
- planar upper bounds distance to planar by a constant – by definition
- graphs with bounded size are not included in graph class planar – By definition
- planar is equivalent to planar – assumed