path

Definition: Vertices can be ordered such that exactly those next to each other in the order are connected by an edge.

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

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 24 : graph class path has unbounded modular-width – The Modular-width of a path $P$ with length $n > 3$ is $n$.
- page 26 : graph class path has unbounded neighborhood diversity – The Neighborhood Diversity of a Path $P$ with length $n > 3$ is $n$.
2010 Comparing 17 graph parameters by Sasák
- page 30 : path upper bounds carving-width by a constant – … path with carving-width 2.
assumed
- path upper bounds connected by a constant – by definition
- path upper bounds linear forest by a constant – by definition
- graph class path is included in graph class tree
- graph class tree is not included in graph class path
- graph class path is included in graph class grid
- graph class grid is not included in graph class path
- path is equivalent to path – assumed
unknown source
- graph class path has unbounded treedepth
- graph class path has unbounded distance to cluster – trivially
- graph class path has unbounded diameter – trivially
- graph class path has unbounded maximum matching