Categorical Constructions in Graphs Laura Scull Fort Lewis College, - - PowerPoint PPT Presentation

Categorical Constructions in Graphs

Laura Scull

Fort Lewis College, Durango, CO

FMCS May 2019

Category of Graphs Products Exponentials Homotopy

Outline

The Category of Graphs Products of Graphs Exponential Object Homotopy

Category of Graphs Products Exponentials Homotopy

The Category of Graphs

Throughout this talk, Gph is the category where:

• Objects are graphs G with:
• A finite set of vertices V (G)
• A finite set of edges E(G) which are unordered pairs of

vertices {v w}

• We have at most one edge between any two vertices; loops are

allowed.

Category of Graphs Products Exponentials Homotopy

The Category of Graphs

Throughout this talk, Gph is the category where:

• Objects are graphs G with:
• A finite set of vertices V (G)
• A finite set of edges E(G) which are unordered pairs of

vertices {v w}

• We have at most one edge between any two vertices; loops are

allowed.

• Morphisms f : G → H maps vertices to vertices and respect

adjacency:

• A set map V (G) → V (H)
• If v

w ∈ E(G), then f (v) f (w) ∈ E(H)

Category of Graphs Products Exponentials Homotopy

The Category of Graphs

In Gph, maps respect adjacency: 1 2 3 4 f

f(1) f(2),f(3) f(4)

Category of Graphs Products Exponentials Homotopy

Graph Morphsims

Let Pn denote the graph: 1 2 3 · · · n

• Map Pn → G:

1 2 3 4 − →

Category of Graphs Products Exponentials Homotopy

Graph Morphsims

Let Pn denote the graph: 1 2 3 · · · n

• Map Pn → G:

1 2 3 4 − →

f (1) f (2) f (4) f (3)

• a list of vertices v1v2v3 . . . vn such that vn

vn+1 ∈ E(G)

• a walk in G

Category of Graphs Products Exponentials Homotopy

Graph Morphisms

Let Kn denote the graph with n vertices, all connected to each

• ther (but not themselves):

1 2 3 4

• Map Kn → G:

1 2 3 4 − →

Category of Graphs Products Exponentials Homotopy

Graph Morphisms

Let Kn denote the graph with n vertices, all connected to each

• ther (but not themselves):
• Map Kn → G:

1 2 3 4 − →

f (1) f (3) f (2) f (4)

• a set of vertices {v1v2v3 . . . vn} which are all connected

pairwise

• a clique in G

Category of Graphs Products Exponentials Homotopy

Graph Morphisms

Let K2 denote the graph with 2 vertices: 1 2

• Map G → K2:

− → 1 2

Category of Graphs Products Exponentials Homotopy

Graph Morphisms

Let K2 denote the graph with 2 vertices: 1 2

• Map G → K2:

− → 1 2

• NOT POSSIBLE!
• each vertex gets assigned a label, either 1 or 2, and is not

connected to anyone of the same label

• a bipartition of G

Category of Graphs Products Exponentials Homotopy

Graph Morphisms

Let Kn denote the graph with n vertices, all connected to each

• ther (but not themselves).
• Map G → Kn:

− → 1 2 3 4

Category of Graphs Products Exponentials Homotopy

Graph Morphisms

Let Kn denote the graph with n vertices, all connected to each

• ther (but not themselves).
• Map G → Kn:

− → 1 2 3 4

• each vertex gets assigned a label (colour), and is not

connected to anyone of the same label

• a (proper) colouring of G

Category of Graphs Products Exponentials Homotopy

Homotopy for Graphs?

Continuous deformation does not make sense for graphs. Previous approach: create a simplicial space associated with a graph, apply homotopy to it.

• Babson and Kosolov
• Dochtermann

To develop homotopy internal to Gph: A homotopy is a map X × I → Y ? A homotopy is a map X → Y I?

Category of Graphs Products Exponentials Homotopy

Graph Products

A (categorical) product X × Y : Y

• X1

X1 × X2

• X2

Excercise: What is the categorical graph product?

Category of Graphs Products Exponentials Homotopy

Graph Products

Definition

The (categorical) product graph G × H (also called the tensor or Kronecker product) is defined by:

• A vertex is a pair (v, w) where v ∈ V (G) and w ∈ V (H).
• There is an edge (v1, w1) (v2, w2) ∈ E(G × H) whenever

v1 v2 ∈ E(G) and w1 w2 ∈ E(H).

Category of Graphs Products Exponentials Homotopy

Graph Products

Example

Let G be given by a single looped vertex x, and H = K2:

x

G

1

H Products: G × G

(x, x)

Category of Graphs Products Exponentials Homotopy

Graph Products

Example

Let G be given by a single looped vertex x, and H = K2:

x

G

1

H Products: G × G

(x, x)

Category of Graphs Products Exponentials Homotopy

Graph Products

Example

Let G be given by a single looped vertex x, and H = K2:

x

G

1

H Products: G × G

(x, x) (x, 0) (x, 1)

G × H

Category of Graphs Products Exponentials Homotopy

Graph Products

Example

Let G be given by a single looped vertex x, and H = K2:

x

G

1

H Products: G × G

(x, x) (x, 0) (x, 1)

G × H

Category of Graphs Products Exponentials Homotopy

Graph Products

Example

Let G be given by a single looped vertex x, and H = K2:

x

G

1

H Products: G × G

(x, x) (x, 0) (x, 1)

G × H

(0, 0) (0, 1) (1, 0) (1, 1)

H × H

Category of Graphs Products Exponentials Homotopy

Graph Products

Example

Let G be given by a single looped vertex x, and H = K2:

x

G

1

H Products: G × G

(x, x) (x, 0) (x, 1)

G × H

(0, 0) (0, 1) (1, 0) (1, 1)

H × H

Category of Graphs Products Exponentials Homotopy

Graph Products

A B

G

4 3 1 2

H

Category of Graphs Products Exponentials Homotopy

Graph Products

A B

G

A1 B1 A2 B2 A3 B3 A4 B4

G × H

4 3 1 2

H

Category of Graphs Products Exponentials Homotopy

Graph Products

A B

G

A1 B1 A2 B2 A3 B3 A4 B4

G × H

4 3 1 2

H

Category of Graphs Products Exponentials Homotopy

Graph Products

This is the categorical product:

A B

G

A1 B1 A2 B2 A3 B3 A4 B4

G × H

4 3 1 2

H

Category of Graphs Products Exponentials Homotopy

π0(G)

• For spaces, π0(X) is the set of connected components.
• For graphs, we can make a similar definition.
• If we exclude isolated vertices, this is a homotopy invariant.
• BUT: it is not product preserving.

1

H

(0, 0) (0, 1) (1, 0) (1, 1)

H × H

Category of Graphs Products Exponentials Homotopy

π0(G) Joint with FLC Senior Capstone Class 2018

Definition

Let G be a graph with no isolated vertices. Define π0(G) as the graph with

• vertices are equivalence classes of vertices in G
• v ∼ w if there is an even length walk from v to w
• there is an edge connecting [v] to [w] when there is an odd

length walk from v to w

Category of Graphs Products Exponentials Homotopy

Example of π0(G)

G =

Category of Graphs Products Exponentials Homotopy

Example of π0(G)

G = π0(G) =

Category of Graphs Products Exponentials Homotopy

Example of π0(G)

4 3 1 2

H

Category of Graphs Products Exponentials Homotopy

Example of π0(G)

4 3 1 2

H π0(G) =

[1] = [3] [2] = [4]

Category of Graphs Products Exponentials Homotopy

π0(G)

Example

G = a1 a2 a3 b1 b2 b3 b4

Category of Graphs Products Exponentials Homotopy

π0(G)

Example

G = a1 a2 a3 b1 b2 b3 b4 π0(G) = A B

Category of Graphs Products Exponentials Homotopy

π0(G)

Results:

• π0 defines a functor from Gph to Gph
• π0 is a homotopy invariant
• π0(G × H) ≃ π0(G) × π0(H)

Category of Graphs Products Exponentials Homotopy

Examples

a b a b

(a, a) (a, b) (b, a) (b, b)

G G G × G

Category of Graphs Products Exponentials Homotopy

Examples

a b a b

(a, a) (a, b) (b, a) (b, b)

G G G × G π0(G) π0(G) π0(G × G)

Category of Graphs Products Exponentials Homotopy

Examples

A B

G

2 3 1

H

Category of Graphs Products Exponentials Homotopy

Examples

A B

G

2 3 1

H

[A] [B]

π0(G)

[1] = [2] = [3]

π0(H)

Category of Graphs Products Exponentials Homotopy

Examples

A B

G

2 3 1

H

A1 B1 A2 B2 A3 B3

G × H

Category of Graphs Products Exponentials Homotopy

Examples

A1 B1 A2 B2 A3 B3

G × H π0(G × H)

Category of Graphs Products Exponentials Homotopy

Examples

A B

G

3 4 1 2

H

Category of Graphs Products Exponentials Homotopy

Examples

A B

G

3 4 1 2

H

[A] [B]

π0(G)

[1] = [3] [2] = [4]

π0(H)

Category of Graphs Products Exponentials Homotopy

Examples

A B

G

3 4 1 2

H

A1 B1 A2 B2 A3 B3 A4 B4

G × H

Category of Graphs Products Exponentials Homotopy

Examples

A1 B1 A2 B2 A3 B3 A4 B4

G × H π0(G × H)

Category of Graphs Products Exponentials Homotopy

Factoring Graphs

We know that the following two products are the same: × = × = To what extent can we find other multiple factorizations?

Category of Graphs Products Exponentials Homotopy

Exponential Objects in Graphs

Recall we had another way of approaching homotopy:

• G → HI where HI is the exponential object

Excercise: What do exponential objects look like in graphs?

Category of Graphs Products Exponentials Homotopy

Exponential Object in Graphs

Theorem (Dochtermann ’09)

For graphs G, H, let HG denote the graph where

• The vertices V (HG) = {f |f : V (G) → V (H)} are set maps of

vertices (NOT graph homomorphisms)

• The edges are defined by fg ∈ E(HG) iff for all vw ∈ E(G),

we have f (v)g(w) ∈ E(H). Then HG is the exponential object. Observe: a vertex f represents a graph homomorphism iff it is looped in HG.

Category of Graphs Products Exponentials Homotopy

Exponential Object in Graphs

G =0 1 H =a b aa ab ba bb HG =

Category of Graphs Products Exponentials Homotopy

Exponential Object in Graphs

G =0 1 H =a b aa ab ba bb HG =

Category of Graphs Products Exponentials Homotopy

Exponential Object in Graphs

G =0 1 H =a b aa ab ba bb HG =

Category of Graphs Products Exponentials Homotopy

Exponential Object in Graphs

G =0 1 H =a b c a b c a b c 1 HG =

Category of Graphs Products Exponentials Homotopy

Exponential Object in Graphs

G =0 1 H =a b c a b c a b c 1 HG =

Category of Graphs Products Exponentials Homotopy

Exponential Object in Graphs

G =0 1 H =a b c a b c a b c 1 HG = Question: What information is this encoding?

Category of Graphs Products Exponentials Homotopy

Homotopy for Graph Maps

Let In denote the graph: 1 2 · · · n Then f ≃ g : G → H if there exists Λ : G × In → H such that Λ(−, 0) = f and Λ(−, 1) = g.

Category of Graphs Products Exponentials Homotopy

Example of Homotopy

1 2 1 2

g(0), g(2)

f (0) g(1) f (1) f (2)

f g

Category of Graphs Products Exponentials Homotopy

Example of Homotopy

To show that f is homotopic to g, we need Λ : G × In → H 1 2 a b

(a, 0) (a, 1) (b, 0) (b, 1) (b, 2) (a, 2)

I1 G G × I1

Category of Graphs Products Exponentials Homotopy

Homotopy in Graphs

Λ defines a homotopy from f to g: 1 2 1 2 G × I1

g(0), g(2)

f (0) g(1) f (1) f (2)

Λ

Category of Graphs Products Exponentials Homotopy

What do homotopies of graphs do?

Definition

Let f , g : G → H. We say that f and g are a spider pair if

• there is x ∈ V (G) such that f (y) = g(y) for all x = y.
• if x

x in G, then f (x) g(x).

Lemma

If f and g are a spider pair, then f ≃ g.

Category of Graphs Products Exponentials Homotopy

Homotopy from Earlier

1 2 1 2

g(0), g(2)

f (0) g(1) f (1) f (2)

f g f and g are a spider pair.

Category of Graphs Products Exponentials Homotopy

More Spider Moves

u v u = ρ(v) ρ The identity map and ρ are a spider pair.

Category of Graphs Products Exponentials Homotopy

More Spider Moves

1 2 3 4 f (1) f (2) f (3) f (4) f 1 2 3 4 g(1) g(2) g(3) g(4) g

Category of Graphs Products Exponentials Homotopy

Spider Lemma

Proposition

If f ≃ g then there is a finite sequence of morphisms f = f0, f1, f2, . . . , fn = g such that each successive pair fk, fk+1 is a spider pair.

Proof.

If f ≃ g via a length 1 homotopy, then we can define fk(vi) =

• f (vi)

for i ≤ n − k g(vi) for i > n − k and check that each successive pair is a spider pair.

Category of Graphs Products Exponentials Homotopy

Decomposition into Spiders

Let G = C4 and H = P2 with vertices labeled as below. 1 2 3 G a b c H With morphisms f , g: a

f (1)

b

f (0), f (2)

c

f (3)

f a

g(3)

b

g(0), g(2)

c

g(1)

g f ≃ g but f , g are not a spider pair.

Category of Graphs Products Exponentials Homotopy

Decomposition into Spiders

We introduce the morphism h. a

f (1)

b

f (0), f (2)

c

f (3)

f a

h(1), h(3)

b

h(0), h(2)

c h a

g(3)

b

g(0), g(2)

c

g(1)

g

Category of Graphs Products Exponentials Homotopy

Homotopy Equivalence

Definition

A map f : G → H is a homotopy equivalence if there is a map g : H → G such that fg ≃ idH and gf ≃ idG. u v u = ρ(v) ρ What graphs are homotopy equivalent?

Category of Graphs Products Exponentials Homotopy

Folds

Definition

If G is a graph, we say that a morphism f : G → G is a fold if f and the identity map are a spider pair. u v u = ρ(v) ρ

Theorem

If f is a fold, then f : G → im(f ) is a homotopy equivalence. Koslov proved this using simplicial complexes. We have a proof internal to graphs.

Category of Graphs Products Exponentials Homotopy

Stiff graphs

Suppose that f : V (G) → V (G) where f is the identity on all vertices except w. Let v = f (w).

Theorem

Then f is a fold if and only if N(w) ⊆ N(v). In the literature, a fold is sometimes called a dismantling.

Definition

A graph is called stiff if for any pair of vertices v = w, it follows that N(w) ⊆ N(v).

Definition

A graph G is homotopy minimal if it is not homotopy equivalent to any proper subgraph of itself.

Theorem

M is homotopy minimal if and only if M is stiff.

Category of Graphs Products Exponentials Homotopy

Stiff graphs

Theorem

Given any finite graph G, there is a unique (up to isomoprhism) homotopy minimal graph M such that G is homotopy equivalent to M.

Proof.

• Key Lemma: If G is stiff, then idG is NOT homotopic to any
• ther endomoprhism.
• Induction.
• Let M, M′ both be homotopy minimal graphs equivalent to G.

Then ∃f : M → M′, g : M′ → M are homotopy equivalences. Then gf = idM, fg = idM′.

Category of Graphs Products Exponentials Homotopy

Stiff graphs

Hell & Nestril have an equivalent result, that every graph folds down to a unique (up to iso) stiff graph.

Category of Graphs Products Exponentials Homotopy

Pleats

Definition (Shoutout to Jeffery Johnson!)

Let the pleat of G denote the (unique up to isomoprhism) homotopy minimal graph which is homotopy equivalent to G. The subcategory of pleats (stiff graphs) gives a skeleton for the homotopy category.

Category of Graphs Products Exponentials Homotopy

Pleats

Some examples of pleats:

Category of Graphs Products Exponentials Homotopy

π1(G)

Define ∆ : Pn+2 → Pn by folding over the end: 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

∆

This is a homotopy equivalence (two folds). We have an induced map on walks: ∆∗(v0v1 . . . , vn−1vn) = (v0v1 . . . vn−1vnvn−1vn).

Definition

Take WG to be the colimit over ∆∗ inclusions. We define Π(G) to be the the set of homotopy classes rel endpoints of the walks of WG.

Category of Graphs Products Exponentials Homotopy

π1(G)

Definition

Let α be a walk of the form: α = (v0v1v2 . . . vi−1vivi+1vivi+3 . . . vn) then the prune of α is α′ = (v0v1v2 . . . vi−1vivi+3 . . . vn)

Theorem

Π(G) consists of equivalence classes of walks in G under:

• pruning
• homotopy rel endpoints

Category of Graphs Products Exponentials Homotopy

π1(G)

Example

α : a c b c e spider move to α′ : a c e c e prune to α′′ : a c e spider move to β : a d e d a e c b d a e c b d a e c b