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.