RELATIONS on GRAPHS & HYPERGRAPHS John Stell School of Computing University of Leeds
RELATIONS on GRAPHS & HYPERGRAPHS
John Stell
School of ComputingUniversity of Leeds
PaddingtonReading
SwanseaSwindon
Bham N.St Bham IntlEuston
Oxford
Reading Paddington
EustonBirmingham
Birmingham
London
Given image A ⊆ Z2 and structuring element
E ⊆ Z2, dilation A⊕E, and erosion AE are:
A⊕ E = {x ∈ Z2 | ∃y ∈ (x+ E∗) · y ∈ A}
A E = {x ∈ Z2 | ∀y ∈ (x+ E) · y ∈ A}
where E∗ = {−e | e ∈ E}.
opening: A ◦ E = (A E)⊕ E,closing: A • E = (A⊕ E) E.
A ◦ E =⋃{x+ E | x+ E ⊆ A}
A • E = X − ⋃{x+ E∗ | x+ E∗ ⊆ (X −A)}
PX
⊕R>
⊥<
RPX
(PX)op
−op
∧
∼= −
∨ R∗ >
⊥<
⊕R∗(PX)op
−op
∧
∼= −
∨
A relation R on a set X is equivalently:
A subset of X ×X
A function X → PX
A sup-preserving function PX → PX
What if we want relations on a graph to cor-
respond to
sup-preserving functions on the lattice of sub-
graphs?
Is there a notion of converse (and symmetry)
for such relations?
Lattice is subgraphs is a bi-Heyting algebra.
Instead of the Boolean complement we have
the pseudocomplement ¬
and its dual or supplement¬
a
b c
d
e
f
s t
u v x
w y
z
a
b c
d
ef
s t
u v
xw y
z
A hypergraph consists of a set H and a re-
lation ϕ on H s.t.
x ϕ y ⇒ (y ϕ z ⇔ y = z).
A sub-hypergraph is K ⊆ H s.t K ⊕ ϕ ⊆ K
A hypergraph relation on (H,ϕ) is a relation
R on H such that R ; ϕ ⊆ R and ϕ ;R ⊆ R.
Hypergraph relations are closed under com-
position, with identity Iϕ = IH ∪ ϕ.
R is a hypergraph relation iff R = Iϕ ;R ; Iϕ
The quantale of hypergraph relations on (H,ϕ)
is isomorphic to the quantale of sup-preserving
mappings on the lattice of sub-hypergraphs.
It is clear what reflexivity and transitivity mean
for hypergraph relations, but is there an ana-
logue of symmetry?
Can we define the converse of a hypergraph
relation?
Lϕ
δ>
⊥<
ε
Lϕ
(Lϕ)op
¬op
∧
a¬
∨ (ε∗)op>
⊥<
(δ∗)op
(Lϕ)op
¬op
∧
` ¬
∨
We can define the converse of δ to be ¬ ;ε ;¬
but what does this mean in terms of rela-
tions?
In fact, it’s the same as taking the converse
of a hypergraph relation R to be Iϕ ; R∗ ; Iϕ
where R∗ is the ordinary converse.
Writing R← = Iϕ ;R∗ ; Iϕ we find
(R ; S)← 6 S← ;R←
R 6 (R←)←
Iϕ 6 (Iϕ)←
Back to the original motivation: granularity
Can we use this notion of converse to de-
fine symmetry and would the corresponding
notion of equivalence relation give a good
notion of partition?
First recall the notion of interior for sub-
graphs.
We can consider R 6 R← as a criterion for
symmetry
When R satisfies this and also R ; R 6 R we
do get:
If x and y are nodes and y⊕R intersects the
interior of x⊕R then y ⊕R 6 x⊕R.