Paths in Graphs and Continua Paul Gartside May 2018 University of - - PowerPoint PPT Presentation

Paths in Graphs and Continua

Paul Gartside May 2018

University of Pittsburgh

Joint work with: Max Pitz, University of Hamburg and Benjamin Espinoza, University of Pittsburgh - Greensburg Ana Mamatelashvili, Melbourne

Building and Crossing Bridges

Problem (Euler) For which graphs G is there a path crossing every edge exactly once?

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Building and Crossing Bridges

Problem (Euler) For which graphs G is there a path crossing every edge exactly once?

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What is a Graph? A Path?

‘Graph’ G: Combinatoric object: vertices V , edges E . . . Topological object: 1-complex. ‘Path’ (from a to b): combinatorially or topologically.

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Euler’s Solution

Theorem (Euler) Let G = (V , E) be a finite graph. Then TFAE: (a) there is a closed path in G crossing each edge exactly once (b) every vertex has even degree, and (c) for every partition, A, B of V the number of edges starting in A and ending in B is even.

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Euler’s Solution

Theorem (Euler) Let G = (V , E) be a finite graph. Then TFAE: (a) there is an open path in G crossing each edge exactly once (b) every vertex except 2 has even degree, and (c) there are 2 vertices a, b such that for every partition, A, B of V with a in A and b in B the number of edges starting in A and ending in B is even.

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Hamilton’s Problem

Problem (Hamilton) For which graphs G is there a path visiting every vertex exactly once? Note: a Hamiltonian open path is, topologically, an arc (∼ = [0, 1]). There is no characterization of Hamiltonian graphs.

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Steps in The Plan

Hamilton Euler Graphs Graph-like Continua

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Citizens! Assert Your Rights!

Note: G = (V , E) open Hamiltonian iff the vertices V are contained in an arc. Definition A graph G is n-open Hamilton iff any points x1, . . . , xn are contained in an arc. Definition A graph G is n-closed Hamilton iff any points x1, . . . , xn are contained in a circle.

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Citizens! Assert Your Rights!

Note: G = (V , E) open Hamiltonian iff the vertices V are contained in an arc. Definition A space X is n-arc connected iff any points x1, . . . , xn are contained in an arc. Definition A space X is n-circle connected iff any points x1, . . . , xn are contained in a circle.

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They will (not) control us

Theorem A graph G is 6-ac if and only if either G is 7-ac or, after suppressing all degree-2-vertices, the graph G is 3-regular, 3-connected, and removing any 6 edges does not disconnect G into 4 or more components.

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7 = ∞

Theorem Let G be a non-degenerate graph. Then the following are equivalent: (a) G is 7-ac, (b) G is n-ac, for all n, and (c) after suppressing or adding degree 2 vertices G is isomorphic to one of 9 graphs.

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Menger

Theorem (Menger’s Theorem) Let G = (V , E) be a (potentially infinite) graph and A, B ⊆ V . Then the minimum number of vertices separating A from B in G is equal to the maximum number of disjoint A − B paths in G. But we need to use algorithmic versions using alternating paths.

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We have. . .

Characterized n-ac and n-cc graphs G for: all n, and all graphs.

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From Finite to Infinite Graphs

Problem How to lift results from finite graphs to infinite graphs.

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From Finite to Infinite Graphs

Problem How to lift results from finite graphs to countable, locally finite graphs.

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Eulerianity

Theorem (Euler) Let G be a finite (connected) graph. Then: G is Eulerian (there is a closed path crossing every edge exactly once) if and only if every vertex is even.

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Eulerianity

Theorem (Euler) Let G be a finite (connected) graph. Then: G is open Eulerian (there is an open path crossing every edge exactly once) if and only if two vertices are odd, the rest even.

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Problems with Infinite Graphs

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4-Regular Tree

Sabidussi: connected graph is Eulerian if every vertex even and just one end.

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Solution: Add the Ends, and Compactify

Diestel. Berger, Bowler, Bruhn, Carmesin, Christian, Georgakopoulos, Richter, Rooney, Stein.

• R. Diestel,

Locally finite graphs with ends: a topological approach I-III, Discrete Math (2010–11).

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Freudenthal Compactification

Adding all ends gives the Freudenthal compactification. Equivalently, maximal compactification with 0-dimensional remainder. Finite graph G Freudenthal compactification γG cycle circle path ‘standard’ path closed path ‘standard’ loop edge-disjoint edge-disjoint Eulerian Eulerian

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Graph-Like Spaces and Continua

Definition (Thomassen and Vella, 2008) A graph-like space (respectively, continuum) is a triple (X, V , E) where: X is a compact, metrizable space (respectively, continuum), V ⊆ X is a closed zero-dimensional subset, and E is a discrete index set such that X \ V ∼ = E × (0, 1). The Freudenthal compactification of a locally finite graph is graph-like.

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Cantor Bouquet of Circles

I CBC

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Characterizing Graph-Like Continua

Theorem The following are equivalent for a continuum X: (i) X is graph-like, (ii) X is completely regular, (iii) X is a countable inverse limit of finite connected multi-graphs with onto, monotone bonding maps that project vertices onto vertices, and (iv) X is homeomorphic to a connected standard subspace of a Freudenthal compactification of a locally finite graph.

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Eulerian Graph-like Continua

Theorem Let X be a graph-like continuum with vertices V . TFAE: (i) X is closed [open] Eulerian, (ii) every vertex is even [apart from precisely two vertices which are

• dd], and

(iii) [there are vertices x = y such that] for every partition of V into two clopen pieces, the number of cross edges is even [if and only if x and y lie in the same part].

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Back (Briefly) to the Hamilton Side

Theorem For the Freudenthal compactification γG of a locally finite connected graph G and for each n ≥ 2: γG is n-ac if and only if G is n-ac. Theorem For every n ≥ 2, there are 2ℵ0 many non-homeomorphic graph-like continua which are n-ac [n-cc] but not (n + 1)-ac [(n + 1)-cc].

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Eulerian Continua

A continuum is Eulerian if it satisfies any of following conditions: Theorem For a continuous surjection f : S1 → X onto X, TFAE: (1) f is arcwise increasing; (2) f is irreducible. (3) f is hereditarily irreducible; (4) f is strongly irreducible; (5) f is almost injective; and Moreover, if X has a dense collection of free arcs E, then also (6) f traverses every edge exactly once, as an embedding, and f −1(E) dense in S1.

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Our Conjecture

Gartside, Pitz A Peano continuum X is Eulerian if and only if X∼ is Eulerian, where X∼ is the graph-like continuum obtained by identifying to points all components of X \

• {all open free arcs}.

This strengthens conjecture of Bula, Nikiel and Tymchatyn.

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Alternative Phrasing

Gartside, Pitz Let X be a Peano continuum X. Write E = {all open free arcs} and V = X \ E. Then X is Eulerian if and only if for every partition of V induced by A, B open in X the number of edges (components of E) from A to B is even. “All edge-cuts even”

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Two Reductions

Theorem (Espinoza, Matsuhashi) Continua without free arcs are Eulerian. So we need only consider continua with free arcs. And wlog these arcs are dense: Theorem Let X be a Peano continuum with free arcs indexed by E, and let D be a countable dense subset for F = X \ E. Let X ′ = X ∪ L be the Peano continuum where we attach a zero-sequence of loops L = {ℓd} : d ∈ D to points in D. Then if X ′ is Eulerian, then so is X.

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Eulerianity conjecture for Peano graphs

Conjecture (Gartside & Pitz): A Peano graph is Eulerian if and only if all its edge-cuts are even. Peano graph: Peano compactification of locally finite, countable graph G via boundary ∂G. Conjecture true – when ∂G = S1.

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Framework: Approximation by finite Eulerian graphs

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Framework: Approximation by finite Eulerian graphs

1 Partition into almost Eulerian tiles. (This step uses Bing’s Brick Partition Theorem and the theory of TST’s, fundamental circuits and thin sums by Diestel et al...).

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Framework: Approximation by finite Eulerian graphs

G1 1 Partition into almost Eulerian tiles. (This step uses Bing’s Brick Partition Theorem and the theory of TST’s, fundamental circuits and thin sums by Diestel et al...). 2 Let G1 be graph on the tiles with edge set all uncovered edges.

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Framework: Approximation by finite Eulerian graphs

G1 3 Carefully add dummy edges to G1 in order to make it Eulerian. (This step uses the assumption that all cuts of X are even).

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Framework: Approximation by finite Eulerian graphs

G1 3 Carefully add dummy edges to G1 in order to make it Eulerian. 4 Add one dummy loop for each new dummy edge at the intersection of corresponding tiles.

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Framework: Approximation by finite Eulerian graphs

G1 3 Carefully add dummy edges to G1 in order to make it Eulerian. 4 Add one dummy loop for each new dummy edge at the intersection of corresponding tiles. 5 Repeat!

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Framework: Approximation by finite Eulerian graphs

G1 1′ Partition each tile into (smaller) almost Eulerian tiles.

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Framework: Approximation by finite Eulerian graphs

G2 2′ Obtain a “finer” graph G2 on the new tiles.

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Framework: Approximation by finite Eulerian graphs

G2 2′ Obtain a “finer” graph G2 on the new tiles. 3′ Add dummy edges to G2 in order to make it Eulerian–inside the old tiles! (For this to be possible, we need that the old tiles were almost Eulerian).

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Framework: Approximation by finite Eulerian graphs

G2 2′ Obtain a “finer” graph G2 on the new tiles. 3′ Add dummy edges to G2 in order to make it Eulerian 4′ Add dummy loop for each new dummy edge at the intersection

• f corresponding tiles.

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Framework: Approximation by finite Eulerian graphs

G2 2′ Obtain a “finer” graph G2 on the new tiles. 3′ Add dummy edges to G2 in order to make it Eulerian 4′ Add dummy loop for each new dummy edge at the intersection

• f corresponding tiles.

5′ Repeat!

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Framework: Approximation by finite Eulerian graphs

G2 ⇒ Obtain a sequence of finite Eulerian graphs G1, G2, G3, . . . such that every Gi can be obtained by edge-contraction from Gi+1. Note: E(T) ⊆ E(Gi).

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Framework: Approximation by finite Eulerian graphs

G2 ⇒ Baire Category: find Eulerian path not using two dummy edges in a row; the restriction to E(T) will be an Eulerian path for the hyperbolic tree.

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Outlook

Conjecture – long way to go. . . Graphs are Spaces – build bridges to graph theory. Compactify Graphs – more bridges. Graph Problems – may translate to continua theory problems.

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