pathwidth

abbr: pw

providers: ISGCI

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

Results

2019 The Graph Parameter Hierarchy by Sorge
- page 11 : treedepth upper bounds pathwidth by a linear function – Lemma 4.27. The treedepth $t$ strictly upper bounds the pathwidth $p$. We have $p \le t$.
2015 Linear rank-width and linear clique-width of trees by Adler, Kanté
- page 3 : pathwidth upper bounds linear rank-width by a computable function – Lemma 5. Any graph $G$ satisfies $\mathrm{lrw}(G) \le \mathrm{pw}(G)$.
2010 Comparing 17 graph parameters by Sasák
- page 16 : graph class tree has unbounded pathwidth – Theorem 2.1
2009 Clique-Width is NP-Complete by Fellows, Rosamond, Rotics, Szeider
- page 8 : pathwidth upper bounds linear clique-width by a linear function – (5) $\mathrm{lin-cwd}(G) \le \mathrm{pwd}(G)+2$.
2005 On the relationship between NLC-width and linear NLC-width by Gurski, Wanke
- page 8 : pathwidth upper bounds linear NLC-width by a linear function – The results of [23] imply that each graph class of bounded path-width has bounded linear NLC-width and that each graph class of bounded tree-width has bounded NLCT-width.
1998 A partial $k$-arboretum of graphs with bounded treewidth by Bodlaender
- page 4 : pathwidth upper bounds treewidth by a linear function – Lemma 3. (a) For all graphs $G$, $pathwidth(G) \ge treewidth(G)$. …
- page 23 : bandwidth upper bounds pathwidth by a linear function – Theorem 44. For every graph $G$, the pathwidth of $G$ is at most the bandwidth of $G$, … Proof. Let $f \colon V\to {1,\dots,n}$ be a linear ordering of $G$ with bandwidth $k$. Then $(X_1,\dots,X_{n-k})$ with $X_i={f^{-1}(i), f^{-1}(i+1), \dots, f^{-1}(i+k)}$ is a path decomposition of $G$ with pathwidth $k$. …
- page 23 : topological bandwidth upper bounds pathwidth by a linear function – Theorem 45. For every graph $G$, the pathwidth of $G$ is at most the topological band-width of $G$.
- page 24 : cutwidth upper bounds pathwidth by a linear function – Theorem 47. For every graph $G$, the pathwidth of $G$ is at most the cutwidth of $G$.
unknown source
- pathwidth upper bounds linear rank-width by a computable function
- distance to linear forest upper bounds pathwidth by a linear function – After removal of $k$ vertices the remaining class has a bounded width $w$. So by including the removed vertices in every bag, we can achieve decomposition of width $w+k$
- distance to linear forest upper bounds pathwidth by a linear function – After removal of $k$ vertices the remaining class has a bounded width $w$. So by including the removed vertices in every bag, we can achieve decomposition of width $w+k$
- cycles upper bounds pathwidth by a constant – trivially
assumed
- pathwidth+maxdegree upper bounds pathwidth by a linear function – by definition
- pathwidth is equivalent to pathwidth – assumed
Comparing Graph Parameters by Schröder
- page 23 : bounded feedback edge set does not imply bounded pathwidth – Proposition 3.16