Nowhere-dense graph classes and algorithmstelle/zdenek.pdfNowhere-dense graph classes and algorithms Z. Dvoˇrák Introduction Subgraph problem Tree-depth Orderings Generalized coloring

Nowhere-dense graphclasses andalgorithms

Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings

Generalizedcoloringnumber

Boundedexpansion

Nowhere-dense graphclasses

Shallowminors

Closures

Orientations

Conclusions

Nowhere-dense graph classes andalgorithms

Z. Dvorák

Geilo winter school


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Credits and advertisements

unless I attribute the results to somebody, they are byNešetril and Ossona de Mendezthey are also preparing a comprehensive book on thesubject:

Sparsity (Graphs, Structures, and Algorithms)


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Plan of the lecture

tell something about nowhere-dense graph classes andgraph classes with bounded expansionand algorithms for thema little about data structureswe will avoid mostly theoretical connections to logic andtheory of homomorphisms


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Warning

The term bounded expansion used in this talk has noimmediate connection to

edge/vertex expansion of graphs, orexpanders.

I apologize for sticking with this somewhat unfortunatename.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Graph classes

a graph class: a set (more precisely, proper class) ofgraphs closed on isomorphismwe only consider finite graphs without loops andparallel edges“nowhere-dense” and “bounded expansion” areproperties of graph classes

not of single graphse.g., the class of all planar graphs has boundedexpansion


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Plan of the lecture

usually, I start by defining what “nowhere-dense” and“bounded expansion” meansbut then lot of time is spent by explaining the definitionsso, let’s start with the algorithms


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Subgraph problem

Problem

Input: graphs H and G.Question: is H a subgraph of G?


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

General algorithms for subgraph problem

NP-complete (H = Kk is a special case).trivial algorithm in O(knk ), where n = |V (G)| andk = |V (H)|.less trivially in nωk/3 (Nešetril and Poljak). Idea:

K3 ⊆ G⇔ E(G2) ∩ E(G) 6= ∅G2 computed by matrix multiplication.

Can f (k)nO(1) algorithm exist (FPT)?unlikely – W [1]-complete.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Restricting G

What if G belongs to some special class of graphs?G has tree-width at most t : f (k , t)O(n).G has maximum degree at most d : f (k ,d)O(n)

G is planar: f (k)O(n) (Eppstein)G does not contain Kt as a minor: f (k , t)nO(1) (Dawar,Grohe and Kreutzer)

Using:localitydecompositions


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Decompositions of graphs

Idea: partition V (G) to a small number of parts, s.t. union ofevery |V (H)| of them induces a graph with simple structure(e.g., bounded tree-width).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Decompositions of graphs

Definition

(p, tw ≤ t)-coloring of G is a coloring such that union ofevery p color classes induces a graph of tree-width atmost t .

Algorithm:find a (k , tw ≤ t(k))-coloring of G by m(k) colorsfor each k color classes C1, . . . , Ck , test whetherH ⊆ G[C1 ∪ . . . ∪ Ck ].

Time complexity O(

c(k ,n) +(m(k)

k

)f (k , t(k))n

), where

c(k ,n) is the complexity of finding the coloringf (k , t)O(n) is the complexity of finding subgraph ingraphs of tree-width at most t .


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Example: planar graphs

Theorem (Robertson and Seymour)

A planar graph of radius r has tree-width at most 3r .

choose a vertex vlet Ci = {u ∈ V (G) : d(u, v) mod (p + 1) = i}

C0, C1, . . . , Cp give a (p, tw ≤ 3p)-coloring by p + 1 colors.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Example: planar graphs

Theorem

For every p, a (p, tw ≤ 3p)-coloring by p + 1 colors can befound in linear time for every planar graph.

Consequently,

Theorem

Testing whether H ⊆ G can be done in O(kf (k ,3k)n) forevery planar graph G.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Proper minor-closed classes

Theorem (DeVos, Ding, Oporowski, Sanders, Reed,Seymour and Vertigan)

If C is a proper minor-closed class of graphs, then for everyp, every G ∈ C has a (p, tw ≤ p − 1)-coloring by fC(p) colors.

implies FPT for subgraph testingbut complicated (based on minor structure theory).

Definition

A class of graphs C has low tree-width colorings if thereexists a function g such that for every p, every G ∈ C has a(p, tw ≤ p − 1)-coloring by g(p) colors.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Motivation

we want a simpler algorithm for finding(p, tw ≤ p − 1)-coloringbut tree-width is still a rather complicated parametercan even simpler class of graphs be used instead?

(2, tw ≤ 1)-coloring . . . acyclic coloringunion of any two color classes induces a forestno bichromatic cycles

star coloringunion of any two color classes induces a star forestno bichromatic P4

needs at most quadratic number of colors wrt. acycliccoloring


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Tree-depth

depth of a rooted tree: maximum number of edges on apath to the rootclosure cl(T ) of a rooted tree T : for each v , add edgesfrom v to all vertices on the path from v to the root


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Tree-depth

Definition

Tree-depth td(G) of a connected graph G is the minimumdepth of a rooted tree T such that G ⊆ cl(T ). Tree-depth ofdisconnected graph is the maximum of the tree-depths of itscomponents.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Properties of tree-depth

1 td(G) = 0 . . . isolated vertices; td(G) = 1 . . . star forest2 minor-monotone3 td(G) ≥ pw(G) ≥ tw(G)

4 G connected: td(G) = 1 + min{td(G − v) : v ∈ V (G)}5 td(Kn) = n − 1, td(Pn) = blog2 nc


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Tree-depth and paths

Theorem

blog2 pc ≤ td(G) ≤(p+1

2

), where p is the number of vertices

of the longest path in G.

Proof.

P ⊆ G is a path on p vertices⇒ G − V (P) does not containany path on p vertices:

td(G) ≤ p + td(G − V (P)) ≤ p +(p

2

)by induction


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Tree-depth coloring

Definition

(p, td ≤ t)-coloring of G is a coloring such that union ofevery p color classes induces a graph of tree-depth atmost t .

Definition

A class of graphs C has low tree-depth colorings if thereexists a function g such that for every p, every G ∈ C has a(p, td ≤ p − 1)-coloring by g(p) colors.

Does any non-trivial graph class have this property?


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Tree-depth versus tree-width

Claim

There exists a function g such that for every t a p, everygraph with tree-width at most t has a(p, td ≤ p − 1)-coloring by g(t ,p) colors.

We will prove a stronger result later. For now:

Corollary

If G has a (p, tw ≤ t)-coloring by c colors, then it also has a(p, td ≤ p − 1)-coloring by at most

cg(t ,p)(cp)

colors.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions


Proof of the Corollary.

Let ϕ be the (p, tw ≤ t)-coloringLet C1, C2, . . . , C(c

p)be all possible unions of p color

classes and let ϕi be a (p, td ≤ p − 1)-coloring of G[Ci ]by at most g(t ,p) colors

and define ϕi arbitrarily on V (G) \ Ci

Assign each vertex v the color(ϕ(v), ϕ1(v), . . . , ϕ(c

p)(v))

any union of at most p color classes in this coloring is asubset of some Ciand thus also a subset of a union of at most p colorclasses of ϕi


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions


Corollary (of the Corollary)

A class of graphs has low tree-width colorings if and only if ithas low tree-depth colorings.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

How to find a coloring?

Greedy algorithm:remove a vertex v of smallest degree, color the rest ofthe graph, then color v by the smallest possible color

Reformulation: let v1, v2, . . . , vn be an ordering of V (G).backdegree of vi is the number of its neighbors amongv1, v2, . . . , vi−1

coloring number of the ordering is the maximum ofbackdegrees of the vertices

Definition

Coloring number col1(G) is the minimum of coloringnumbers of all possible orderings of V (G).

Note: χ(G) ≤ col1(G) + 1.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

What about acyclic coloring?

Arrangeability: let v1, v2, . . . , vn be an ordering of V (G).vj is 2-backreachable from vi if j < i and there exists apath P of length at most two between vi and vj , suchthat the internal vertex vm (if any) of P satisfies i < m.2-backdegree of v is the number of vertices2-backreachable from varrangeability of the ordering is the maximum of2-backdegrees of vertices

Definition

Arrangeability col2(G) is the minimum of arrangeabilities ofall possible orderings of V (G).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Arrangeability


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

What about acyclic coloring?

Theorem

G has an acyclic coloring by at most col2(G) + 1 colors.

Proof.

color vertices in the order certifying the arrangeability,assign colors different from 2-backreachable verticesno bichromatic cycle:


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Generalized coloring number

Let v1, v2, . . . , vn be an ordering of V (G).an s-backpath from vi to vj with j < i is a path of lengthat most s such that if vm is an internal vertex of P, theni < mvj is s-backreachable from vi if there exists ans-backpath from vi to vj

the s-backdegree of v is the number of verticess-backreachable from vthe s-coloring number of the ordering is the maximumof s-backdegrees of the vertices

Definition

The s-coloring number cols(G) is the minimum of s-coloringnumbers of all possible orderings of V (G).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Working with generalized coloring number

Problems:Does generalized coloring number give us lowtree-depth colorings?How to determine it (and find the ordering)?

NP-complete.How to approximate it?


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Iterated backreachability

Let v1, v2, . . . , vn be an ordering of V (G).

Definition

va is (s, r)-backreachable from vb, if there exist indicesa = i0, i1, . . . , it = b, where t ≤ r , and vij is s-backreachablefrom vij+1 for 0 ≤ j < t .

If the ordering has s-coloring number d , then at mostd + d2 + . . .+ d r < (d + 1)r vertices are(s, r)-backreachable from any vertex.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

col→ low tree-depth colorings

Theorem

Every graph has (p, td ≤ p − 1)-coloring by at most(cols(G) + 1)s colors, where s = 2p−1.

Proof.

colors different from (s, s)-backreachable verticesunion of every t ≤ p color classes has td ≤ t − 1:


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

How to determine s-coloring number?

greedy algorithmchoose vertices vn, vn−1, . . .always pick a vertex with smallest s-backdegree

problem: picking vi may increase s-backdegrees ofremaining vertices

it is possible to make a wrong choice

solution: minimize a different parameter


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Admissibility

Let v1, v2, . . . , vn be an ordering of V (G).the s-backconnectivity of a vertex vi is the maximumnumber of s-backpaths from vi that intersect only in vi

the s-admissibility of the ordering is the maximum ofthe s-backconnectivities of the vertices

Definition

The s-admissibility adms(G) is the minimum ofs-admissibilities of all possible orderings of V (G).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Admissibility


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Admissibility

Observation: greedy algorithm correctly determinesadms(G)


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Remarks on algorithm for admissibility

Problem: determining s-backconnectivity is NP-complete fors ≥ 5.

but, testing whether it is less than a given constant is inP, andcan be approximated within the factor of s (greedily)

Testing whether adms(G) ≤ a for fixed a and s can beimplemented in O(n) using further results.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Admissibility vs coloring number

Theorem

Let v1, v2, . . . , vn be an ordering of V (G), c its s-coloringnumber and a its s-admissibility. Then a ≤ c ≤ as.

Proof.

let T be the tree of shortest s-backpaths from vi

∆(T ) ≤ ahence, T has at most as leaves


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Bounded admissibilities

Definition

A class of graphs C has bounded admissibilities if thereexists a function f such that for every s and every G ∈ C,adms(G) ≤ f (s).

So far, we proved the following.

Theorem (Zhu)

Any class of graphs with bounded admissibilities has lowtree-depth colorings.

Which graph classes have bounded admissibilities?


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Subdivisions

For a graph G,the k -subdivision sdk (G) is the graph created from Gby subdividing every edge by exactly k verticesa (≤k)-subdivision is a graph created from G bysubdividing every edge by at most k vertices

not necessarily every edge the same number of times;some edges may remain unsubdivided

a (≤k)-topological minor of G is any H such that some(≤k)-subdivision of H is a subgraph of G.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Admissibility of subdivisions

Theorem

If H is a (≤s − 1)-subdivision of a graph with minimumdegree d ≥ 3, then adms(H) ≥ d.

Proof.

let v be the last vertex of degree at least three in theorderingthe s-backconnectivity of v is at least d


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Subdivisions in high-admissibility graphs

Theorem

If adms(G) > (16d)s, then G has a (≤s − 1)-topologicalminor H such that δ(H) > d.

Proof.

Otherwise, average degree of any (≤s − 1)-topologicalminor is ≤ 2d .Greedy algorithm fails, with set M of unchosen vertices:


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions


We have more than (16d)s|M| paths.Almost gives the topological minor, but the paths mayoverlap.We need to clean up the paths; consecutively by levels(s − 1 times)Assuming levels up to i consist of disjoint paths:

1 throw away paths ending with the next step: −4d |M|2 if next step of P ends in level j < i of Q, throw away P

or Q: /33 for each vertex reachable in level i + 1, choose one of

the paths: /4d

The resulting graph is too dense.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions



Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Summary

Definition

Let ∇s(G) be the largest minimum degree of an(≤s)-topological minor of G.

We have

∇s−1(G) ≤ adms(G) ≤ (16∇s−1(G))s.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Definition

Definition

A class of graphs C has bounded expansion if there exists afunction f such that for every s and every G ∈ C,∇s(G) ≤ f (s).

Theorem (Zhu; D.)

A class has bounded expansion if and only if it has boundedadmissibilities.

Recall: bounded admissibilities⇒ low tree-depth colorings.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Low tree-depth colorings and subdivisions

Lemma

If δ(H) > d, then every subdivision H ′ of H has td(H ′) > d.

Proof.

Consider the vertex v of degree greater than d appearingdeepest in the tree certifying tree-depth of H ′:


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Low tree-depth colorings and subdivisions

Theorem

If sds(G) has an (s + 2, td ≤ s + 1) coloring by at most ccolors, then δ(G) ≤ 2(s + 1)

( cs+2

).

Proof.

For e ∈ E(G), let Se be the set of colors on thecorresponding path (including endvertices).For S ⊆ {1, . . . , c} of size s + 2, let GS be the subgraphwith edges {e : Se ⊆ S}.For some S, GS has average degree at leastδ(G)/

( cs+2

), and contains G′ ⊆ GS with

δ(G′) ≥ δ(G)/[2( c

s+2

)].

Observe td(sds(G′)) ≤ s + 1 and apply lemma.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Low tree-depth colorings and boundedexpansion

Corollary

If a class of graphs has low tree-depth colorings, then it hasbounded expansion.

I.e., the following are equivalent:bounded expansionbounded admissibilitieshaving low tree-depth coloringshaving low tree-width colorings


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Classes with bounded expansion

Theorem

Any proper class C of graphs closed on topological minorshas bounded expansion.

Proof.

Kk 6∈ C for some kif H ∈ C, then δ(H) ≤ O(k2) (Komlós)closed on topological minors: ∇s(G) ≤ O(k2) for everyG ∈ C and s ≥ 0.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Classes with bounded expansion

Corollary

The following graph classes have bounded expansion:graphs with bounded maximum degreeproper minor-closed graph classes, e.g.,

graphs with bounded tree-widthplanar graphs


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Remark on bounded tree-width

Bounded tree-width⇒ bounded expansion⇒ lowtree-depth colorings, proving

Claim

There exists a function g such that for every t a p, everygraph with tree-width at most t has a(p, td ≤ p − 1)-coloring by g(t ,p) colors.

as we promised before.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Other classes with bounded expansion

graphs drawn in a fixed surface with a bounded numberof crossings on each edgecreated by adding edges in mutual distance ω(1) tographs in any class with bounded expansionalmost all graphs with linear number of edges


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Generalizations of the subgraph problem

For all graph classes with bounded expansion (D., Král’,Thomas):

testing first-order properties in linear timee.g., having dominating set of size at most k (k fixed):

(∃x1) . . . (∃xk )(∀y) y = x1 ∨ . . . ∨ y = xk ∨E(y , x1) ∨ . . . ∨ E(y , xk ).

data structure for graphs with colored vertices andedges

linear-time initializationchange color of an element in O(1)decide first-order query with bounded number ofquantifiers in O(1)


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Nowhere-dense graph classes

Definition

A class of graphs C is nowhere-dense if there exists afunction f such that for every s, Kf (s) is not an(≤s)-topological minor of any graph in C.

Equivalently, for every s, the set of (≤s)-topologicalminors of graphs in C does not contain all graphs.Bounded expansion⇒ nowhere-dense


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Properties of nowhere-dense classes

If C is nowhere-dense, then for every ε > 0, integer s andG ∈ C with n vertices:

∇s(G) = O(nε), henceadms(G) = O(nε), henceG has (s, td ≤ s − 1) coloring by O(nε) colors, hencewe can test H ⊆ G (for fixed H) in O(n1+ε).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Other results

Most results for graph classes with bounded expansion alsoholds for nowhere-dense graph classes (with O(nε)replacing constants). Exception:

Problem

Are first-order properties FPT on nowhere-dense graphclasses?


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Subgraph problem and nowhere-dense graphclasses

Theorem

Assume that the subgraph problem is not FPT on the classof all graphs. If the subgraph problem is FPT on a class ofgraphs C closed on subgraphs, then C is nowhere-dense.

Proof.

For every s ≥ 0,

H ⊆ G⇔ sds(H) ⊆ sds(G).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Examples of nowhere-dense classes

locally bounded expansion, includinglocally bounded tree-widthlocally proper minor closed

Definition

A class of graphs C has locally bounded expansion if thereexists a function f such that for every s,d ≥ 0 and everyG ∈ C, if H is a subgraph of G of radius at most d , then∇s(H) ≤ f (s,d).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Nowhere-dense 6= bounded expansion

The class

C = {G : ∆(G) ≤ log log |V (G)|,girth(G) ≥ log log |V (G)|}

is nowhere-dense: if sds(Kk ) ∈ C, thenk − 1 ≤ log log |V (G)| ≤ 3s.does not have bounded expansion: unboundedminimum degree


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Original definition of bounded expansion

Bounded expansion has many differentcharacterizations.But we still did not see the one that came the firstchronologically.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Shallow minors

Definition

A depth r minor of G is a graph obtained from a subgraph ofG by contracting vertex-disjoint subgraphs of radius at mostr .


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Shallow minors

Definition

Let ∇r (G) be the greatest average density |E(H)|/|V (H) ofa depth r minor H of G.

Remark: Greatest Reduced Average Density, hence the ∇symbol.

Theorem (D.)

For any r ≥ 0 and any graph G,

∇2r (G) ≤ 2∇r (G) ≤ 4(4∇2r )(r+1)2.

Proof.

Idea: split the spanning trees of the shallow minor onvertices of big enough degree.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Shallow minors and bounded expansion

Corollary

A class of graphs C has bounded expansion if and only ifthere exists a function f such that for every r and everyG ∈ C, ∇r (G) ≤ f (r).

We say that the expansion of C or of the graph G is boundedby f .


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Small separators

Definition

A set S ⊆ V (G) is a separator if each component of G − Shas at most 2|V (G)|/3 vertices.

Theorem (Plotkin, Rao and Smith)

If a graph G on n vertices does not contain Kh as depthd log2 n minor, then G has a separator of size at mostO(n/d + dh2 log n). Can be found in O(|E(G)|n/d).

Corollary

If there exists c ≥ 0 such that the expansion of G is boundedby O(r c), then G has a separator of size (n log n)1−1/(2c+2).If the expansion of G is bounded by a subexponentialfunction, then G has separator of sublinear size. Tightbecause of 3-regular expanders.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Consequences of small separators

Corollary

If the expansion of G is bounded by a subexponentialfunction, then G has sublinear tree-width.

Corollary (D., Norine)

For any function f such that lim supr→∞log log f (r)

log r < 1/3,there exists c > 0 such that the number of non-isomorphicgraphs G on n vertices with expansion bounded by f is atmost cn.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Blowing up vertices

Lemma

Let H be the graph obtained from G by blowing up eachvertex to a clique of size k. Then ∇s(H) is bounded bya function of ∇s(G) and k.

Proof.

Let F ′ ⊆ H be an (≤s)-subdivision of a graph F withδ(F ) = ∇s(H). Each e ∈ E(F ) has a path Pe ⊂ F ′.

Pe does not contain twins, unless it is an edgemerge twin branchpoints: min. degree ≥ (δ − k + 1)/kremove twins of branchpoints: average degreeA ≥ (δ − k + 1)/k − 2(k − 1)

Each Pe now conflicts with ≤ (k − 1)s other paths. Chooselargest subgraph where all paths are independent.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Blowing up vertices


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Contracting a forest

Lemma

Let H be the graph obtained from G by contracting a forest.Then ∇s(H) is bounded by a function of ∇3s+5(G).

Proof.

∇s(H) ≤ 2∇ds/2e(H) ≤ 2∇3ds/2e+1(G) ≤ 2f (∇3s+5(G))


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Application

Theorem

For any fixed k ≥ 0, the class of graphs that can be drawn inplane with at most k crossings on each edge has boundedexpansion.

Proof.

Put vertices on crossings and subdivide the edges: planargraph, with bounded expansion. Blow up all vertices tocliques of size two, contract forest: creates crossings.Suppress vertices of degree two (at most 2k contractions ofa forest).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Orientations with bounded degree

Claim

∇0(G) ≤ d if and only if G has an orientation with indegreeat most d.

compare with: if ∇0(G) ≤ d , then G is 2d-degenerate(the reverse implication does not hold)

equivalently, G has an acyclic orientation with indegree≤ 2d

we will now consider orientations whose acyclicversions correspond to generalized coloring numbers


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Augmentations

Let G be a directed graph. An unordered pair {u, v} isa transitive pair if uw ,wv ∈ E(G) for some w ∈ V (G)

a fraternal pair if uw , vw ∈ E(G) for some w ∈ V (G)

Definition

A directed graph H is an augmentation of G if the edge setof the underlying undirected graph of H consists of theedges of G and of all transitive and fraternal pairs in G.

I.e., add the transitive and fraternal pairs as edges and givethem arbitrary orientations.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Density of augmentations

Theorem

If H is the underlying undirected graph of an augmentationof G and d is the maximum indegree of G, then ∇s(H) isbounded by a function of d and ∇3s+5(G).

Proof.

H is a subgraph of graph obtained by replacing each vertexby d + 1 vertices and contracting a star forest:

In particular, there exists an augmentation of G withmaximum indegree bounded by a function of d and ∇5(G).We call such augmentation steady.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Steady augmentations

Theorem

If H is the underlying undirected graph of an augmentationof G and d is the maximum indegree of G, then ∇s(H) isbounded by a function of d and ∇3s+5(G).

In particular, there exists an augmentation of G withmaximum indegree bounded by a function of d and ∇5(G).We call such an augmentation steady.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Short paths via augmentations

Lemma

Let s ≥ 2, let G be a graph, G0 its orientation and G0, G1,G2, . . . , Gs a sequence of augmentations. If the distancebetween u and v in G is at most (3/2)s−1 + 2, then u and vare either adjacent or have a common in-neighbor in Gs.

Proof.

In each augmentation, the path of length t gives rise to apath of length ≤ 2/3t + 2/3:


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Oracle for short paths

Fix s ≥ 2. Find an orientation G0 of G with boundedindegree and compute augmentations G1, . . . , Gs.

for each edge, remember the length of the shortestcorresponding pathplus one of the ways how it was createdfor bounded expansion classes, if steadyaugmentations are used:

linear-time preprocessingby Lemma, O(1) queries for paths of length at most(3/2)s−1 + 2 (every vertex has only O(1) in-neighbors).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Dynamic version

idea: maintain the augmentations when edges areadded or removedproblem: adding edge can create unbounded numberof edges due to transitive pairssolution: only add edges for fraternal pairs (fraternalaugmentation).


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Short paths via fraternal augmentations

Lemma

Let s ≥ 0, let G be a graph, G0 its orientation and G0, G1,G2, . . . , Gs a sequence of fraternal augmentations. If thedistance between u and v in G is at most s + 1, then thereexists a path P = w1w2 . . .wt between u = w1 and v = wt inGs of length at most s + 1, and an index c ≤ t such that theedges w1w2, w2w3, . . . , wc−1wc are oriented towards u andthe rest of the edges is oriented towards v.

To find the path, search in-neighbors up to distance s + 1(O(1) if steady augmentations are used)


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Maintaining the orientation

Theorem (Brodal and Fagerberg)

For any d > 0 there exists D so that an orientation of ad-degenerate graph on n vertices with maximum indegreeD can be maintained within the following time bounds:

an edge can be added in O(log n) (amortized)an edge can be removed in O(1)

Each vertex stores a list of in- and out-neighbors.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Maintaining the augmentations

adding an edge results in O(log n) reorientations in G0

each reorientation adds or removes O(1) edges in G1

O(log2 n) reorientations in G1, . . .

Gives O(logs n) update time for maintaining paths of lengthat most s.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Low tree-depth colorings via augmentations

Theorem

Let s ≥ 1, let G be a graph, G0 its orientation and G0, G1,G2, . . . , Gp a sequence of augmentations, wherep = 3(s + 1)2. Let H be the underlying undirected graph ofGp. Then any proper coloring of H gives an(s, td ≤ s − 1)-coloring of G.

For bounded expansion classes, linear-time and the numberof colors is bounded, if steady augmentations are used.Proof is lengthy and technical.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions


Theorem

Let s ≥ 1, let G be a graph, G0 its orientation and G0, G1,G2, . . . , Gp a sequence of augmentations, where p =

(s+12

).

Any proper coloring of Gp is an(

s, td ≤(2s

2

))-coloring of G.

Proof.

Show that no path on 2s vertices uses ≤ s colors, sincethe subgraph induced by the path contains Ks+1.After

(s2

)augmentations we have two disjoint Ks;

have directed Hamiltonian paths, starts v1 and v2,v1 and v2 adjacent or a common in-neighbor.Say v1 has an in-neighbor w outside of its clique,next s augmentations to add w to the clique.


Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions



Z. Dvorák

Introduction

Subgraphproblem

Tree-depth

Orderings


Boundedexpansion


Shallowminors

Closures

Orientations

Conclusions

Summary

Bounded expansion and nowhere-dense serve as goodformalization of “structurally sparse” graphs.Many natural graph classes have these properties.Problems expressible in first-order logic can be solvedefficiently for them.Many other results for special classes of sparse graphsgeneralize to this setting.

Documents

Nowhere-dense graph classes and algorithmstelle/zdenek.pdfNowhere-dense graph classes and algorithms Z. Dvoˇrák Introduction Subgraph problem Tree-depth Orderings Generalized coloring