iterated type partitions

tags: module

Definition: see Parameterized complexity for iterated type partitions and modular-width by Cordasco, Gargano, Rescigno

Relations

Other		Relation from	Relation to
acyclic chromatic number	blue■	exclusion	exclusion
admissibility	blue■	exclusion	exclusion
arboricity	blue■	exclusion	exclusion
average degree	blue■	exclusion	exclusion
average distance	red■	exclusion	upper bound
bandwidth	cyan■	unknown to HOPS	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	blue■	exclusion	exclusion
boolean width	red■	exclusion	upper bound
bounded components	cyan■	unknown to HOPS	exclusion
bounded expansion	blue■	exclusion	avoids
boxicity	magenta■	exclusion	unknown to HOPS
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	blue■	exclusion	exclusion
chi-bounded	red■	exclusion	upper bound
chordal	blue■	unbounded	exclusion
chordality	magenta■	exclusion	unknown to HOPS
chromatic number	blue■	exclusion	exclusion
clique cover number	blue■	exclusion	exclusion
clique-tree-width	magenta■	exclusion	unknown to HOPS
clique-width	red■	exclusion	upper bound
cluster	cyan■	unknown to HOPS	exclusion
co-cluster	cyan■	unknown to HOPS	exclusion
cograph	cyan■	unknown to HOPS	exclusion
complete	green■	upper bound	exclusion
connected	blue■	exclusion	avoids
contraction complexity	blue■	exclusion	exclusion
cutwidth	blue■	exclusion	exclusion
cycle	cyan■	unknown to HOPS	exclusion
cycles	cyan■	unknown to HOPS	exclusion
d-admissibility	magenta■	exclusion	unknown to HOPS
d-path-free	cyan■	unknown to HOPS	exclusion
degeneracy	blue■	exclusion	exclusion
degree treewidth	blue■	exclusion	exclusion
diameter	red■	exclusion	upper bound
diameter+max degree	cyan■	unknown to HOPS	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	cyan■	unknown to HOPS	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	green■	upper bound	exclusion
distance to edgeless	green■	upper bound	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	cyan■	unknown to HOPS	exclusion
domatic number	blue■	exclusion	exclusion
domination number	blue■	exclusion	exclusion
domino treewidth	blue■	exclusion	exclusion
edge clique cover number	green■	upper bound	exclusion
edge connectivity	blue■	exclusion	exclusion
edge-cut width	blue■	exclusion	exclusion
edge-treewidth	blue■	exclusion	exclusion
edgeless	green■	upper bound	avoids
excluded minor	blue■	exclusion	avoids
excluded planar minor	cyan■	unknown to HOPS	avoids
excluded top-minor	blue■	exclusion	avoids
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	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	yellow■	equal	equal
linear clique-width	magenta■	exclusion	unknown to HOPS
linear forest	blue■	unbounded	exclusion
linear NLC-width	magenta■	exclusion	unknown to HOPS
linear rank-width	magenta■	exclusion	unknown to HOPS
maximum clique	blue■	exclusion	exclusion
maximum degree	blue■	exclusion	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	magenta■	exclusion	unknown to HOPS
maximum leaf number	cyan■	unknown to HOPS	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
merge-width	red■	exclusion	upper bound
mim-width	red■	exclusion	upper bound
minimum degree	blue■	exclusion	exclusion
mm-width	blue■	exclusion	exclusion
modular-width	orange■	unknown to HOPS	upper bound
module-width	red■	exclusion	upper bound
monadically dependent	red■	exclusion	upper bound
monadically stable	magenta■	exclusion	unknown to HOPS
neighborhood diversity	green■	upper bound	exclusion
NLC-width	red■	exclusion	upper bound
NLCT-width	magenta■	exclusion	unknown to HOPS
nowhere dense	magenta■	exclusion	unknown to HOPS
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	cyan■	unknown to HOPS	exclusion
overlap treewidth	blue■	exclusion	exclusion
path	blue■	unbounded	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	blue■	exclusion	exclusion
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	blue■	exclusion	exclusion
sparse twin-width	blue■	exclusion	exclusion
star	green■	upper bound	exclusion
stars	cyan■	unknown to HOPS	exclusion
strong coloring number	blue■	exclusion	exclusion
strong d-coloring number	magenta■	exclusion	unknown to HOPS
strong inf-coloring number	blue■	exclusion	exclusion
topological bandwidth	cyan■	unknown to HOPS	exclusion
tree	blue■	unbounded	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	cyan■	unknown to HOPS	exclusion
treelength	red■	exclusion	upper bound
treespan	blue■	exclusion	exclusion
treewidth	blue■	exclusion	exclusion
twin-cover number	cyan■	unknown to HOPS	exclusion
twin-width	red■	exclusion	upper bound
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	cyan■	unknown to HOPS	exclusion
weak coloring number	blue■	exclusion	exclusion
weak d-coloring number	magenta■	exclusion	unknown to HOPS
weak inf-coloring number	cyan■	unknown to HOPS	exclusion
weakly sparse	magenta■	exclusion	unknown to HOPS
weakly sparse and merge width	blue■	exclusion	exclusion

Results

2024 Parameterized complexity for iterated type partitions and modular-width by Cordasco, Gargano, Rescigno
- page 3 : iterated type partitions – two nodes have the same type iff $N(v) \setminus {u} = N(u) \setminus {v}$ … [ed. paraphrased] let $\mathcal V = {V_1,\dots,V_t}$ be a partition of graph vertices such that each $V_i$ is a clique or an independent set and $t$ is minimized … we can see each element of $\mathcal V$ as a \emph{metavertex} of a new graph $H$, called \emph{type graph} of $G$ … We say that $G$ is a \emph{prime graph} if it matches its type graph … let $H^{(0)}=G$ and $H^{(i)}$ denote the type graph of $H^{(i-1)}$, for $i \ge 1$. Let $d$ be the smallest integer such that $H^{(d)}$ is a \emph{prime graph}. The \emph{iterated type partition} number of $G$, denoted by $\mathrm{itp}(G)$, is the number of nodes of $H^{(d)}$.
- page 3 : neighborhood diversity upper bounds iterated type partitions by a linear function – … $itp(G) \le nd(G)$. Actually $itp(G)$ can be arbitrarily smaller than $nd(G)$.
- page 3 : graph classes with bounded iterated type partitions are not bounded neighborhood diversity – … $itp(G) \le nd(G)$. Actually $itp(G)$ can be arbitrarily smaller than $nd(G)$.
- page 3 : iterated type partitions upper bounds modular-width by a linear function – By definition
assumed
- iterated type partitions is equivalent to iterated type partitions – assumed