degree treewidth

functionally equivalent to: contraction complexity, domino treewidth, treespan

Definition: an intersetcion of maximum degree, and treewidth

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	blue■	exclusion	exclusion
bandwidth	lime■	upper bound	unknown to HOPS
bipartite	blue■	unbounded	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	magenta■	exclusion	unknown to HOPS
block	blue■	unbounded	exclusion
book thickness	red■	exclusion	upper bound
boolean width	red■	exclusion	upper bound
bounded components	green■	upper bound	exclusion
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	lime■	upper bound	unknown to HOPS
chi-bounded	red■	exclusion	upper bound
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■	exclusion	avoids
contraction complexity	yellow■	upper bound	upper bound
cutwidth	lime■	upper bound	unknown to HOPS
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-admissibility	red■	exclusion	upper bound
d-path-free	blue■	exclusion	exclusion
degeneracy	red■	exclusion	upper bound
degree treewidth	yellow■	equal	equal
diameter	blue■	exclusion	exclusion
diameter+max degree	green■	upper bound	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	blue■	exclusion	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 edgeless	blue■	exclusion	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	blue■	exclusion	exclusion
distance to maximum degree	red■	exclusion	upper bound
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	blue■	exclusion	exclusion
domatic number	red■	exclusion	upper bound
domination number	blue■	exclusion	exclusion
domino treewidth	yellow■	upper bound	upper bound
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	red■	exclusion	upper bound
edge-cut width	magenta■	exclusion	unknown to HOPS
edge-treewidth	red■	exclusion	upper bound
edgeless	green■	upper bound	avoids
excluded minor	magenta■	exclusion	unknown to HOPS
excluded planar minor	gray■	unknown to HOPS	unknown to HOPS
excluded top-minor	red■	exclusion	upper bound
feedback edge set	blue■	exclusion	exclusion
feedback vertex set	blue■	exclusion	exclusion
flip-width	red■	exclusion	upper bound
forest	blue■	unbounded	exclusion
genus	blue■	exclusion	exclusion
grid	blue■	unbounded	exclusion
h-index	red■	exclusion	upper bound
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	green■	upper bound	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	red■	exclusion	upper bound
maximum degree	red■	exclusion	upper bound
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	blue■	exclusion	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	blue■	exclusion	exclusion
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	blue■	exclusion	exclusion
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	red■	exclusion	upper bound
NLCT-width	red■	exclusion	upper bound
nowhere dense	red■	exclusion	upper bound
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	red■	exclusion	upper bound
path	green■	upper bound	exclusion
pathwidth	magenta■	exclusion	unknown to HOPS
pathwidth+maxdegree	lime■	upper bound	unknown to HOPS
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	magenta■	exclusion	unknown to HOPS
sim-width	red■	exclusion	upper bound
size	green■	upper bound	exclusion
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	gray■	unknown to HOPS	unknown to HOPS
tree	blue■	unbounded	exclusion
tree-cut width	red■	exclusion	upper bound
tree-independence number	red■	exclusion	upper bound
tree-partition-width	red■	exclusion	non-tight bounds
treebandwidth	red■	exclusion	upper bound
treedepth	blue■	exclusion	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	yellow■	upper bound	upper bound
treewidth	red■	exclusion	upper bound
twin-cover number	blue■	exclusion	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	magenta■	exclusion	unknown to HOPS
vertex cover	blue■	exclusion	exclusion
vertex integrity	blue■	exclusion	exclusion
weak coloring number	red■	exclusion	upper bound
weak d-coloring number	red■	exclusion	upper bound
weak inf-coloring number	blue■	exclusion	exclusion
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	red■	exclusion	upper bound

Results

2025 On a tree-based variant of bandwidth and forbidding simple topological minors by Jacob, Lochet, Paul
- page 37 : degree treewidth upper bounds domino treewidth by a linear function – Theorem 53. … $\max({\rm tw}(G),(\Delta(G)-1)/2) \le {\rm dtw}(G)$ … [this result is claimed to be in A note on domino treewidth by Bodlaender, didn’t see it there]
- page 37 : treespan upper bounds degree treewidth by a linear function – Claim 55. $\max({\rm tw}(G),\Delta(G)/2) \le {\rm ts}(G)$
2024 Slim Tree-Cut Width by Ganian, Korchemna
- page 2716 : degree treewidth upper bounds slim tree-cut width by a computable function – Figure 1
2009 On tree-partition-width by Wood
- page 2 : degree treewidth upper bounds tree-partition-width by a computable function and lower bounds it by a polynomial function – $twp(G) \le 24tw(G)\Delta(G)$
- page 2 : degree treewidth upper bounds tree-partition-width by a computable function and lower bounds it by a polynomial function – Theorem 1. … $twp(G) < \frac 52 (tw(G)+1)(\frac 72 \Delta-1)$
- page 2 : degree treewidth upper bounds tree-partition-width by a computable function and lower bounds it by a polynomial function – Theorem 2. … there is a chordal graph $G$ … $twp(G) \ge (\frac 18 - \epsilon)tw(G)\Delta(G).$
2008 Simulating Quantum Computation by Contracting Tensor Networks by Markov, Shi
- page 11 : degree treewidth upper bounds contraction complexity by a polynomial function – Proposition 4.2. … $cc(G)=tw(G^)$ … Lemma 4.4. $(tw(G) - 1)/2 \le tw(G^) \le \Delta(G)(tw(G) + 1) - 1.$
1999 A note on domino treewidth by Bodlaender
- page 4 : degree treewidth upper bounds domino treewidth by a linear function – Theorem 3.1 Let $G=(V,E)$ be a graph with treewidth at most $k$ and maximum degree at most $d$. Then the domino treewidth of $G$ is at most $(9k+7)d(d+1)-1$.
- page 7 : degree treewidth upper bounds domino treewidth by a linear function – Lemma 4.3 For all $d \ge 5$, $k \ge 2$, $k$ even, there exists a graph $G$ with treewidth at most $k$, maximum degree at most $d$, and domino treewidth at least $\frac{1}{12} kd-2$.
assumed
- degree treewidth upper bounds maximum degree by a linear function – by definition
- degree treewidth upper bounds treewidth by a linear function – by definition
- degree treewidth is equivalent to degree treewidth – assumed