interval

providers: ISGCI

Definition: Intersection graph of intervals on the real line.

Relations

Other		Relation from	Relation to
acyclic chromatic number	blue■	exclusion	unbounded
arboricity	blue■	exclusion	unbounded
average degree	blue■	exclusion	unbounded
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	magenta■	exclusion	unknown to HOPS
book thickness	blue■	exclusion	unbounded
boolean width	magenta■	exclusion	unknown to HOPS
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	red■	exclusion	inclusion
chordality	red■	exclusion	upper bound
chromatic number	blue■	exclusion	unbounded
clique cover number	blue■	exclusion	unbounded
clique-tree-width	magenta■	exclusion	unknown to HOPS
clique-width	magenta■	exclusion	unknown to HOPS
cluster	green■	inclusion	exclusion
co-cluster	blue■	exclusion	exclusion
cograph	blue■	exclusion	exclusion
complete	green■	upper bound	exclusion
connected	blue■	exclusion	exclusion
contraction complexity	blue■	exclusion	unbounded
cutwidth	blue■	exclusion	unbounded
cycle	cyan■	unknown to HOPS	exclusion
cycles	blue■	exclusion	exclusion
d-path-free	blue■	exclusion	unbounded
degeneracy	blue■	exclusion	unbounded
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	blue■	exclusion	exclusion
distance to bipartite	blue■	exclusion	unbounded
distance to block	magenta■	exclusion	unknown to HOPS
distance to bounded components	blue■	exclusion	unbounded
distance to chordal	red■	exclusion	upper bound
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	magenta■	exclusion	unknown to HOPS
distance to edgeless	blue■	exclusion	unbounded
distance to forest	blue■	exclusion	unbounded
distance to interval	red■	exclusion	upper bound
distance to linear forest	blue■	exclusion	unbounded
distance to maximum degree	blue■	exclusion	unbounded
distance to outerplanar	blue■	exclusion	unbounded
distance to perfect	red■	exclusion	upper bound
distance to planar	blue■	exclusion	unbounded
distance to stars	blue■	exclusion	unbounded
domatic number	blue■	exclusion	unbounded
domination number	blue■	exclusion	unbounded
edge clique cover number	blue■	exclusion	unbounded
edge connectivity	blue■	exclusion	unbounded
edgeless	green■	upper bound	exclusion
feedback edge set	blue■	exclusion	unbounded
feedback vertex set	blue■	exclusion	unbounded
forest	blue■	exclusion	exclusion
genus	blue■	exclusion	unbounded
girth	magenta■	exclusion	unknown to HOPS
grid	blue■	exclusion	exclusion
h-index	blue■	exclusion	unbounded
inf-flip-width	magenta■	exclusion	unknown to HOPS
interval	yellow■	equal	equal
iterated type partitions	blue■	exclusion	unbounded
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	green■	inclusion	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	blue■	exclusion	unbounded
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	blue■	exclusion	unbounded
mm-width	blue■	exclusion	unbounded
modular-width	blue■	exclusion	unbounded
module-width	magenta■	exclusion	unknown to HOPS
neighborhood diversity	blue■	exclusion	unbounded
NLC-width	magenta■	exclusion	unknown to HOPS
NLCT-width	magenta■	exclusion	unknown to HOPS
odd cycle transversal	blue■	exclusion	unbounded
outerplanar	blue■	exclusion	exclusion
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	unbounded
pathwidth+maxdegree	blue■	exclusion	unbounded
perfect	red■	exclusion	upper bound
planar	blue■	exclusion	exclusion
radius-r flip-width	magenta■	exclusion	unknown to HOPS
rank-width	magenta■	exclusion	unknown to HOPS
shrub-depth	magenta■	exclusion	unknown to HOPS
sim-width	magenta■	exclusion	unknown to HOPS
size	blue■	exclusion	unbounded
star	green■	upper bound	exclusion
stars	green■	inclusion	exclusion
topological bandwidth	blue■	exclusion	unbounded
tree	cyan■	unknown to HOPS	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	magenta■	exclusion	unknown to HOPS
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	blue■	exclusion	unbounded
vertex integrity	blue■	exclusion	unbounded

Results

2017 Graph Theory by Diestel
- page 145 : interval – A graph $G$ is called an \emph{interval graph} if there exists a set ${ I_v \mid v \in V(G) }$ of real intervals such that $I_u \cap I_v \ne \emptyset$ if and only if $uv \in E(G)$.
assumed
- graph class cluster is included in graph class interval
- graph class interval is not included in graph class cluster
- graph class linear forest is included in graph class interval
- graph class interval is not included in graph class linear forest
- graph class stars is included in graph class interval
- graph class interval is not included in graph class stars
- graph class interval is included in graph class chordal
- graph class chordal is not included in graph class interval
- interval upper bounds distance to interval by a constant – by definition
- interval is equivalent to interval – assumed
unknown source
- graph class interval has unbounded average distance