maximum degree

providers: ISGCI

Definition: Maximum degree over graph’s vertices.

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	magenta■	exclusion	unknown to HOPS
boolean width	blue■	exclusion	exclusion
bounded components	green■	upper bound	exclusion
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	exclusion
c-closure	red■	exclusion	upper bound
carving-width	green■	upper bound	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	blue■	exclusion	exclusion
clique-width	blue■	exclusion	exclusion
cluster	blue■	unbounded	exclusion
co-cluster	blue■	unbounded	exclusion
cograph	blue■	unbounded	exclusion
complete	blue■	unbounded	exclusion
connected	blue■	unbounded	exclusion
contraction complexity	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-path-free	blue■	exclusion	exclusion
degeneracy	red■	exclusion	upper bound
degree treewidth	green■	upper bound	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	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 disconnected	red■	exclusion	upper bound
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
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	green■	upper bound	exclusion
h-index	red■	exclusion	upper bound
inf-flip-width	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	blue■	exclusion	exclusion
linear forest	green■	upper bound	exclusion
linear NLC-width	blue■	exclusion	exclusion
linear rank-width	blue■	exclusion	exclusion
maximum clique	red■	exclusion	upper bound
maximum degree	yellow■	equal	equal
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
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	red■	exclusion	upper bound
mm-width	blue■	exclusion	exclusion
modular-width	blue■	exclusion	exclusion
module-width	blue■	exclusion	exclusion
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	blue■	exclusion	exclusion
NLCT-width	blue■	exclusion	exclusion
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	blue■	unbounded	exclusion
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	blue■	unbounded	exclusion
planar	blue■	unbounded	exclusion
radius-r flip-width	magenta■	exclusion	unknown to HOPS
rank-width	blue■	exclusion	exclusion
shrub-depth	blue■	exclusion	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
star	blue■	unbounded	exclusion
stars	blue■	unbounded	exclusion
topological bandwidth	cyan■	unknown to HOPS	exclusion
tree	blue■	unbounded	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
treedepth	blue■	exclusion	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treewidth	blue■	exclusion	exclusion
twin-cover number	blue■	exclusion	exclusion
twin-width	blue■	exclusion	exclusion
vertex connectivity	red■	exclusion	upper bound
vertex cover	blue■	exclusion	exclusion
vertex integrity	blue■	exclusion	exclusion

Results

2022 Expanding the Graph Parameter Hierarchy by Tran
- page 32 : maximum degree upper bounds c-closure by a computable function – Proposition 5.1. Maximum Degree strictly upper bounds $c$-Closure.
- page 32 : bounded c-closure does not imply bounded maximum degree – Proposition 5.1. Maximum Degree strictly upper bounds $c$-Closure.
- page 40 : bounded twin-width does not imply bounded maximum degree – Proposition 6.8. Twin-width is incomparable to Maximum Degree.
- page 40 : bounded maximum degree does not imply bounded twin-width – Proposition 6.8. Twin-width is incomparable to Maximum Degree.
2019 The Graph Parameter Hierarchy by Sorge
- page 8 : maximum degree upper bounds acyclic chromatic number by a polynomial function – Lemma 4.6 ([15]). The acyclic chromatic number $\chi_a$ is uppre bounded by the maximum degree $\Delta$ (for every graph with $\Delta > 4$). We have $\chi_a \le \Delta(\Delta-1)/2$.
2013 Characterizing graphs of small carving-width by Belmonte, van ’t Hof, Kamiński, Paulusma, Thilikos
- carving-width upper bounds maximum degree by a linear function – Observation 1. Let $G$ be a graph. Then $cw(G) \ge \Delta(G)$.
2010 Comparing 17 graph parameters by Sasák
- page 28 : cutwidth upper bounds maximum degree by a linear function – Lemma 2.18. For any graph $G$ and any vertex $v \in V(G), cutw(g) \ge \lceil \frac{deg(v)}2 \rceil$.
- page 30 : carving-width upper bounds maximum degree by a linear function – Lemma 2.20 Carving-width of a graph $G$ is at least $\Delta(G)$ where $\Delta(G)$ is the maximum degree of a graph $G$.
2008 Simulating Quantum Computation by Contracting Tensor Networks by Markov, Shi
- page 10 : contraction complexity upper bounds maximum degree by a linear function – $cc(G) \ge \Delta(G) - 1$
unknown source
- maximum degree upper bounds h-index by a linear function – As h-index seeks $k$ vertices of degree $k$ it is trivially upper bound by maximum degree.
- bandwidth upper bounds maximum degree by a linear function – Each vertex has an integer $i$ and may be connected only to vertices whose difference from $i$ is at most $k$. There are at most $k$ bigger and $k$ smaller such neighbors.
- maximum degree upper bounds c-closure by a computable function
- grid upper bounds maximum degree by a constant
- graph class planar has unbounded maximum degree
- graph class star has unbounded maximum degree – trivially
Comparing Graph Parameters by Schröder
- page 24 : bounded vertex cover does not imply bounded maximum degree – Proposition 3.19
- page 28 : bounded maximum degree does not imply bounded clique-width – Proposition 3.26
- page 28 : bounded maximum degree does not imply bounded bisection bandwidth – Proposition 3.26
assumed
- pathwidth+maxdegree upper bounds maximum degree by a linear function – by definition
- degree treewidth upper bounds maximum degree by a linear function – by definition
- maximum degree upper bounds distance to maximum degree by a linear function – by definition
- diameter+max degree upper bounds maximum degree by a linear function – by definition
- grid upper bounds maximum degree by a constant – By definition
- bounded components upper bounds maximum degree by a linear function – By definition
- linear forest upper bounds maximum degree by a constant – By definition
- cycles upper bounds maximum degree by a constant – By definition
- maximum degree is equivalent to maximum degree – assumed