genus

providers: ISGCI

Definition: Genus is the minimum integer $k$ such that the graph can be embedded on a surface with $k$ handles without edge crossings.

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	blue■	exclusion	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	red■	exclusion	upper bound
boolean width	blue■	exclusion	exclusion
bounded components	cyan■	unknown to HOPS	exclusion
bounded expansion	red■	exclusion	upper bound
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chi-bounded	magenta■	exclusion	unknown to HOPS
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■	exclusion	avoids
contraction complexity	blue■	exclusion	exclusion
cutwidth	blue■	exclusion	exclusion
cycle	green■	upper bound	exclusion
cycles	cyan■	unknown to HOPS	exclusion
d-admissibility	red■	exclusion	upper bound
d-path-free	blue■	exclusion	exclusion
degeneracy	red■	exclusion	upper bound
degree treewidth	blue■	exclusion	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	cyan■	unknown to HOPS	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	blue■	exclusion	exclusion
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	blue■	exclusion	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	red■	exclusion	upper bound
edge-cut width	cyan■	unknown to HOPS	exclusion
edge-treewidth	blue■	exclusion	exclusion
edgeless	green■	upper bound	avoids
excluded minor	orange■	unknown to HOPS	upper bound
excluded planar minor	cyan■	unknown to HOPS	avoids
excluded top-minor	red■	exclusion	upper bound
feedback edge set	green■	upper bound	exclusion
feedback vertex set	blue■	exclusion	exclusion
flip-width	red■	exclusion	upper bound
forest	green■	upper bound	exclusion
genus	yellow■	equal	equal
grid	green■	upper bound	exclusion
h-index	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	blue■	exclusion	exclusion
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	cyan■	unknown to HOPS	exclusion
merge-width	red■	exclusion	upper bound
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
monadically dependent	red■	exclusion	upper bound
monadically stable	red■	exclusion	upper bound
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	blue■	exclusion	exclusion
NLCT-width	blue■	exclusion	exclusion
nowhere dense	red■	exclusion	upper bound
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	green■	upper bound	exclusion
overlap treewidth	blue■	exclusion	exclusion
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	blue■	exclusion	exclusion
perfect	blue■	unbounded	exclusion
planar	green■	upper bound	exclusion
radius-inf flip-width	blue■	exclusion	exclusion
radius-r flip-width	red■	exclusion	upper bound
rank-width	blue■	exclusion	exclusion
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	blue■	exclusion	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
slim tree-cut width	blue■	exclusion	exclusion
sparse twin-width	red■	exclusion	upper bound
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
strong coloring number	red■	exclusion	upper bound
strong d-coloring number	red■	exclusion	upper bound
strong inf-coloring number	blue■	exclusion	exclusion
topological bandwidth	blue■	exclusion	exclusion
tree	green■	upper bound	exclusion
tree-cut width	blue■	exclusion	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
tree-partition-width	blue■	exclusion	exclusion
treebandwidth	blue■	exclusion	exclusion
treedepth	blue■	exclusion	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	blue■	exclusion	exclusion
treewidth	blue■	exclusion	exclusion
twin-cover number	blue■	exclusion	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	gray■	unknown to HOPS	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

2024 Twin-width of graphs on surfaces by Kráľ, Pekárková, Štorgel
- page 18 : genus upper bounds twin-width by a linear function – The twin-width of every graph $G$ of Euler genus $g \ge 1$ is at most … $18 \sqrt{47g}+O(1)$.
2022 Expanding the Graph Parameter Hierarchy by Tran
- page 34 : graph classes with bounded c-closure are not bounded genus – Proposition 5.6. $c$-Closure is incomparable to Genus.
- page 34 : graph classes with bounded genus are not bounded c-closure – Proposition 5.6. $c$-Closure is incomparable to Genus.
- page 37 : genus upper bounds twin-width by a linear function – Proposition 6.3. Genus strictly upper bounds Twin-width.
- page 37 : graph classes with bounded twin-width are not bounded genus – Proposition 6.3. Genus strictly upper bounds Twin-width.
2019 The Graph Parameter Hierarchy by Sorge
- page 8 : genus upper bounds acyclic chromatic number by a linear function – Lemma 4.8 ([3]). The accylic chromatic number $\chi_a$ is upper bounded by the genus $g$. We have $\chi_a \le 4g+4$.
- page 10 : feedback edge set upper bounds genus by a linear function – Lemma 4.19. The feedback edge set number $f$ upper bounds the genus $g$. We have $g \le f$.
1994 Genus $g$ Graphs Have Pagenumber $O(\sqrt g)$ by Malitz
- page 24 : genus upper bounds book thickness by a linear function – Theorem 5.1. Genus $g$ graphs have pagenumber $O(\sqrt{g})$.
assumed
- genus is equivalent to genus – assumed
unknown source
- feedback edge set upper bounds genus by a linear function – Removing $k$ edges creates a forest that is embeddable into the plane. We now add one handle for each of the $k$ edges to get embedding into $k$-handle genus.
- planar upper bounds genus by a constant
- genus upper bounds chromatic number by a constant – in fact, bounded by square root
- genus upper bounds excluded minor by a computable function
Comparing Graph Parameters by Schröder
- page 23 : graph classes with bounded genus are not bounded clique-width – Proposition 3.17
- page 24 : graph classes with bounded vertex cover are not bounded genus – Proposition 3.18
- page 26 : graph classes with bounded genus are not bounded distance to perfect – Proposition 3.24
- page 30 : graph classes with bounded bandwidth are not bounded genus – Proposition 3.27
- page 33 : graph classes with bounded genus are not bounded distance to planar – Proposition 3.34