Nowhere-dense graph classes and algorithms by sdfgsg234

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									  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   Nowhere-dense graph classes and
Subgraph
problem                  algorithms
Tree-depth

Orderings

Generalized
coloring
                                  r
                            Z. Dvoˇák
number

Bounded
expansion

Nowhere-                Geilo winter school
dense graph
classes

Shallow
minors

Closures

Orientations
               Credits and advertisements

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph           unless I attribute the results to somebody, they are by
problem
                         r
                   Nešetˇil and Ossona de Mendez
Tree-depth

Orderings
                   they are also preparing a comprehensive book on the
Generalized
                   subject:
coloring
number
                       Sparsity (Graphs, Structures, and Algorithms)
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Plan of the lecture

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction
                   tell something about nowhere-dense graph classes and
Subgraph
problem            graph classes with bounded expansion
Tree-depth         and algorithms for them
Orderings
                   a little about data structures
Generalized
coloring
number
                   we will avoid mostly theoretical connections to logic and
Bounded            theory of homomorphisms
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Warning

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction
               The term bounded expansion used in this talk has no
Subgraph
problem        immediate connection to
Tree-depth         edge/vertex expansion of graphs, or
Orderings
                   expanders.
Generalized
coloring
number
               I apologize for sticking with this somewhat unfortunate
Bounded        name.
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Graph classes

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
                  a graph class: a set (more precisely, proper class) of
Introduction
                  graphs closed on isomorphism
Subgraph
problem           we only consider finite graphs without loops and
Tree-depth        parallel edges
Orderings
                  “nowhere-dense” and “bounded expansion” are
Generalized
coloring          properties of graph classes
number
                      not of single graphs
Bounded
expansion             e.g., the class of all planar graphs has bounded
Nowhere-              expansion
dense graph
classes

Shallow
minors

Closures

Orientations
               Plan of the lecture

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem
                   usually, I start by defining what “nowhere-dense” and
Tree-depth         “bounded expansion” means
Orderings          but then lot of time is spent by explaining the definitions
Generalized
coloring           so, let’s start with the algorithms
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Subgraph problem

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem        Problem
Tree-depth
               Input: graphs H and G.
Orderings

Generalized
               Question: is H a subgraph of G?
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               General algorithms for subgraph problem

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
                   NP-complete (H = Kk is a special case).
Introduction

Subgraph           trivial algorithm in O(knk ), where n = |V (G)| and
problem
                   k = |V (H)|.
Tree-depth

Orderings
                   less trivially in nωk /3 (Nešetˇil and Poljak). Idea:
                                                  r
Generalized             K3 ⊆ G ⇔ E(G2 ) ∩ E(G) = ∅
coloring
number
                        G2 computed by matrix multiplication.
Bounded            Can f (k )nO(1) algorithm exist (FPT)?
expansion
                        unlikely – W [1]-complete.
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Restricting G

  Nowhere-
 dense graph
 classes and
  algorithms

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

Tree-depth         G is planar: f (k )O(n) (Eppstein)
Orderings          G does not contain Kt as a minor: f (k , t)nO(1) (Dawar,
Generalized
coloring
                   Grohe and Kreutzer)
number
               Using:
Bounded
expansion          locality
Nowhere-
dense graph        decompositions
classes

Shallow
minors

Closures

Orientations
               Decompositions of graphs

  Nowhere-
 dense graph
 classes and
  algorithms
               Idea: partition V (G) to a small number of parts, s.t. union of
  Z. Dvoˇák
        r
               every |V (H)| of them induces a graph with simple structure
Introduction   (e.g., bounded tree-width).
Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Decompositions of graphs

  Nowhere-
 dense graph
 classes and   Definition
  algorithms

  Z. Dvoˇák
        r
               (p, tw ≤ t)-coloring of G is a coloring such that union of
               every p color classes induces a graph of tree-width at
Introduction
               most t.
Subgraph
problem
               Algorithm:
Tree-depth

Orderings          find a (k , tw ≤ t(k ))-coloring of G by m(k ) colors
Generalized
coloring
                   for each k color classes C1 , . . . , Ck , test whether
number             H ⊆ G[C1 ∪ . . . ∪ Ck ].
Bounded
                                                m(k )
expansion      Time complexity O c(k , n) +      k      f (k , t(k ))n , where
Nowhere-
dense graph
classes            c(k , n) is the complexity of finding the coloring
Shallow
minors
                   f (k , t)O(n) is the complexity of finding subgraph in
Closures
                   graphs of tree-width at most t.
Orientations
               Example: planar graphs

  Nowhere-
 dense graph   Theorem (Robertson and Seymour)
 classes and
  algorithms
               A planar graph of radius r has tree-width at most 3r .
        r
  Z. Dvoˇák

Introduction        choose a vertex v
Subgraph
problem             let Ci = {u ∈ V (G) : d(u, v ) mod (p + 1) = i}
Tree-depth
               C0 , C1 , . . . , Cp give a (p, tw ≤ 3p)-coloring by p + 1 colors.
Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Example: planar graphs

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Theorem
Introduction

Subgraph
               For every p, a (p, tw ≤ 3p)-coloring by p + 1 colors can be
problem        found in linear time for every planar graph.
Tree-depth

Orderings      Consequently,
Generalized
coloring
number
               Theorem
Bounded        Testing whether H ⊆ G can be done in O(kf (k , 3k )n) for
expansion

Nowhere-
               every planar graph G.
dense graph
classes

Shallow
minors

Closures

Orientations
               Proper minor-closed classes

  Nowhere-
 dense graph
 classes and
  algorithms   Theorem (DeVos, Ding, Oporowski, Sanders, Reed,
  Z. Dvoˇák
        r      Seymour and Vertigan)
Introduction   If C is a proper minor-closed class of graphs, then for every
Subgraph
problem
               p, every G ∈ C has a (p, tw ≤ p − 1)-coloring by fC (p) colors.
Tree-depth

Orderings          implies FPT for subgraph testing
Generalized
coloring
                   but complicated (based on minor structure theory).
number

Bounded
expansion      Definition
Nowhere-
dense graph
               A class of graphs C has low tree-width colorings if there
classes        exists a function g such that for every p, every G ∈ C has a
Shallow
minors
               (p, tw ≤ p − 1)-coloring by g(p) colors.
Closures

Orientations
               Motivation

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r
                   we want a simpler algorithm for finding
                   (p, tw ≤ p − 1)-coloring
Introduction

Subgraph
                   but tree-width is still a rather complicated parameter
problem
                   can even simpler class of graphs be used instead?
Tree-depth
                       (2, tw ≤ 1)-coloring . . . acyclic coloring
Orderings
                            union of any two color classes induces a forest
Generalized
coloring
                            no bichromatic cycles
number                 star coloring
Bounded
expansion
                            union of any two color classes induces a star forest
                            no bichromatic P4
Nowhere-
dense graph                 needs at most quadratic number of colors wrt. acyclic
classes                     coloring
Shallow
minors

Closures

Orientations
               Tree-depth

  Nowhere-
 dense graph
 classes and
  algorithms
                   depth of a rooted tree: maximum number of edges on a
        r
  Z. Dvoˇák
                   path to the root
Introduction
                   closure cl(T ) of a rooted tree T : for each v , add edges
Subgraph
problem            from v to all vertices on the path from v to the root
Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Tree-depth

  Nowhere-
 dense graph
 classes and
  algorithms
               Definition
        r
  Z. Dvoˇák
               Tree-depth td(G) of a connected graph G is the minimum
Introduction

Subgraph
               depth of a rooted tree T such that G ⊆ cl(T ). Tree-depth of
problem        disconnected graph is the maximum of the tree-depths of its
Tree-depth     components.
Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Properties of tree-depth

  Nowhere-
 dense graph
 classes and
  algorithms
                 1   td(G) = 0 . . . isolated vertices; td(G) = 1 . . . star forest
        r
  Z. Dvoˇák
                 2   minor-monotone
Introduction
                 3   td(G) ≥ pw(G) ≥ tw(G)
Subgraph
problem
                 4   G connected: td(G) = 1 + min{td(G − v ) : v ∈ V (G)}
Tree-depth

Orderings
                 5   td(Kn ) = n − 1, td(Pn ) = log2 n
Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Tree-depth and paths

  Nowhere-
 dense graph
 classes and   Theorem
  algorithms

        r
  Z. Dvoˇák
                log2 p ≤ td(G) ≤ p+1 , where p is the number of vertices
                                      2
               of the longest path in G.
Introduction

Subgraph
problem        Proof.
Tree-depth
               P ⊆ G is a path on p vertices ⇒ G − V (P) does not contain
Orderings
               any path on p vertices:
Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors                                           p
Closures
               td(G) ≤ p + td(G − V (P)) ≤ p +   2   by induction
Orientations
               Tree-depth coloring

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r
               Definition
Introduction
               (p, td ≤ t)-coloring of G is a coloring such that union of
Subgraph
               every p color classes induces a graph of tree-depth at
problem
               most t.
Tree-depth

Orderings
               Definition
Generalized
coloring
number
               A class of graphs C has low tree-depth colorings if there
Bounded        exists a function g such that for every p, every G ∈ C has a
expansion
               (p, td ≤ p − 1)-coloring by g(p) colors.
Nowhere-
dense graph
classes

Shallow
               Does any non-trivial graph class have this property?
minors

Closures

Orientations
               Tree-depth versus tree-width

  Nowhere-
 dense graph
 classes and
  algorithms
               Claim
  Z. Dvoˇák
        r      There exists a function g such that for every t a p, every
Introduction
               graph with tree-width at most t has a
Subgraph
               (p, td ≤ p − 1)-coloring by g(t, p) colors.
problem

Tree-depth     We will prove a stronger result later. For now:
Orderings

Generalized    Corollary
coloring
number
               If G has a (p, tw ≤ t)-coloring by c colors, then it also has a
Bounded
expansion      (p, td ≤ p − 1)-coloring by at most
Nowhere-
                                                  c
                                         cg(t, p)(p)
dense graph
classes

Shallow
minors
               colors.
Closures

Orientations
               Tree-depth versus tree-width

  Nowhere-
 dense graph
 classes and   Proof of the Corollary.
  algorithms

  Z. Dvoˇák
        r          Let ϕ be the (p, tw ≤ t)-coloring
Introduction       Let C1 , C2 , . . . , C(c ) be all possible unions of p color
                                         p
Subgraph
problem            classes and let ϕi be a (p, td ≤ p − 1)-coloring of G[Ci ]
Tree-depth         by at most g(t, p) colors
Orderings               and define ϕi arbitrarily on V (G) \ Ci
Generalized        Assign each vertex v the color
coloring
number
                    ϕ(v ), ϕ1 (v ), . . . , ϕ(c ) (v )
Bounded                                      p
expansion               any union of at most p color classes in this coloring is a
Nowhere-                subset of some Ci
dense graph
classes                 and thus also a subset of a union of at most p color
Shallow                 classes of ϕi
minors

Closures

Orientations
               Tree-depth versus tree-width

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem        Corollary (of the Corollary)
Tree-depth

Orderings
               A class of graphs has low tree-width colorings if and only if it
Generalized
               has low tree-depth colorings.
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               How to find a coloring?

  Nowhere-
 dense graph
 classes and
               Greedy algorithm:
  algorithms
                    remove a vertex v of smallest degree, color the rest of
        r
  Z. Dvoˇák
                    the graph, then color v by the smallest possible color
Introduction
               Reformulation: let v1 , v2 , . . . , vn be an ordering of V (G).
Subgraph
problem             backdegree of vi is the number of its neighbors among
Tree-depth
                    v1 , v2 , . . . , vi−1
Orderings

Generalized
                    coloring number of the ordering is the maximum of
coloring
number
                    backdegrees of the vertices
Bounded
expansion      Definition
Nowhere-
dense graph    Coloring number col1 (G) is the minimum of coloring
classes
               numbers of all possible orderings of V (G).
Shallow
minors

Closures
               Note: χ(G) ≤ col1 (G) + 1.
Orientations
               What about acyclic coloring?

  Nowhere-
 dense graph
 classes and
  algorithms   Arrangeability: let v1 , v2 , . . . , vn be an ordering of V (G).
  Z. Dvoˇák
        r           vj is 2-backreachable from vi if j < i and there exists a
Introduction
                    path P of length at most two between vi and vj , such
Subgraph            that the internal vertex vm (if any) of P satisfies i < m.
problem

Tree-depth
                    2-backdegree of v is the number of vertices
Orderings
                    2-backreachable from v
Generalized         arrangeability of the ordering is the maximum of
coloring
number              2-backdegrees of vertices
Bounded
expansion
               Definition
Nowhere-
dense graph
classes
               Arrangeability col2 (G) is the minimum of arrangeabilities of
Shallow        all possible orderings of V (G).
minors

Closures

Orientations
               Arrangeability

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               What about acyclic coloring?

  Nowhere-
 dense graph
 classes and   Theorem
  algorithms

  Z. Dvoˇák
        r
               G has an acyclic coloring by at most col2 (G) + 1 colors.
Introduction

Subgraph
               Proof.
problem

Tree-depth         color vertices in the order certifying the arrangeability,
Orderings          assign colors different from 2-backreachable vertices
Generalized
coloring           no bichromatic cycle:
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Generalized coloring number

  Nowhere-
 dense graph   Let v1 , v2 , . . . , vn be an ordering of V (G).
 classes and
  algorithms        an s-backpath from vi to vj with j < i is a path of length
  Z. Dvoˇák
        r           at most s such that if vm is an internal vertex of P, then
Introduction        i <m
Subgraph
problem
                    vj is s-backreachable from vi if there exists an
Tree-depth
                    s-backpath from vi to vj
Orderings           the s-backdegree of v is the number of vertices
Generalized         s-backreachable from v
coloring
number
                    the s-coloring number of the ordering is the maximum
Bounded
expansion           of s-backdegrees of the vertices
Nowhere-
dense graph
classes
               Definition
Shallow
minors
               The s-coloring number cols (G) is the minimum of s-coloring
Closures
               numbers of all possible orderings of V (G).
Orientations
               Working with generalized coloring number

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction
               Problems:
Subgraph
problem            Does generalized coloring number give us low
Tree-depth         tree-depth colorings?
Orderings
                   How to determine it (and find the ordering)?
Generalized
coloring               NP-complete.
number
                       How to approximate it?
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Iterated backreachability

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r      Let v1 , v2 , . . . , vn be an ordering of V (G).
Introduction

Subgraph
               Definition
problem
               va is (s, r )-backreachable from vb , if there exist indices
Tree-depth
               a = i0 , i1 , . . . , it = b, where t ≤ r , and vij is s-backreachable
Orderings

Generalized
               from vij+1 for 0 ≤ j < t.
coloring
number
               If the ordering has s-coloring number d, then at most
Bounded
expansion      d + d 2 + . . . + d r < (d + 1)r vertices are
Nowhere-
dense graph
               (s, r )-backreachable from any vertex.
classes

Shallow
minors

Closures

Orientations
               col → low tree-depth colorings

  Nowhere-
 dense graph   Theorem
 classes and
  algorithms
               Every graph has (p, td ≤ p − 1)-coloring by at most
  Z. Dvoˇák
        r
               (cols (G) + 1)s colors, where s = 2p−1 .
Introduction

Subgraph       Proof.
problem

Tree-depth
                   colors different from (s, s)-backreachable vertices
Orderings

Generalized
                   union of every t ≤ p color classes has td ≤ t − 1:
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               How to determine s-coloring number?

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction       greedy algorithm
Subgraph               choose vertices vn , vn−1 , . . .
problem
                       always pick a vertex with smallest s-backdegree
Tree-depth

Orderings
                   problem: picking vi may increase s-backdegrees of
Generalized
                   remaining vertices
coloring
number
                       it is possible to make a wrong choice
Bounded            solution: minimize a different parameter
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Admissibility

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r      Let v1 , v2 , . . . , vn be an ordering of V (G).
Introduction        the s-backconnectivity of a vertex vi is the maximum
Subgraph
problem
                    number of s-backpaths from vi that intersect only in vi
Tree-depth          the s-admissibility of the ordering is the maximum of
Orderings           the s-backconnectivities of the vertices
Generalized
coloring
number
               Definition
Bounded
expansion
               The s-admissibility adms (G) is the minimum of
Nowhere-
               s-admissibilities of all possible orderings of V (G).
dense graph
classes

Shallow
minors

Closures

Orientations
               Admissibility

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Admissibility

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Observation: greedy algorithm correctly determines
Introduction
               adms (G)
Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Remarks on algorithm for admissibility

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

               Problem: determining s-backconnectivity is NP-complete for
Introduction

Subgraph
               s ≥ 5.
problem
                   but, testing whether it is less than a given constant is in
Tree-depth
                   P, and
Orderings

Generalized        can be approximated within the factor of s (greedily)
coloring
number
               Testing whether adms (G) ≤ a for fixed a and s can be
Bounded
expansion      implemented in O(n) using further results.
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Admissibility vs coloring number

  Nowhere-
 dense graph   Theorem
 classes and
  algorithms
               Let v1 , v2 , . . . , vn be an ordering of V (G), c its s-coloring
  Z. Dvoˇák
        r
               number and a its s-admissibility. Then a ≤ c ≤ as .
Introduction

Subgraph       Proof.
problem

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

Generalized
                    ∆(T ) ≤ a
coloring
number              hence, T has at most as leaves
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Bounded admissibilities

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r
               Definition

Introduction
               A class of graphs C has bounded admissibilities if there
Subgraph
               exists a function f such that for every s and every G ∈ C,
problem
               adms (G) ≤ f (s).
Tree-depth

Orderings      So far, we proved the following.
Generalized
coloring
number         Theorem (Zhu)
Bounded
expansion      Any class of graphs with bounded admissibilities has low
Nowhere-       tree-depth colorings.
dense graph
classes

Shallow
               Which graph classes have bounded admissibilities?
minors

Closures

Orientations
               Subdivisions

  Nowhere-
 dense graph   For a graph G,
 classes and
  algorithms       the k -subdivision sdk (G) is the graph created from G
  Z. Dvoˇák
        r          by subdividing every edge by exactly k vertices
Introduction       a (≤ k )-subdivision is a graph created from G by
Subgraph           subdividing every edge by at most k vertices
problem
                       not necessarily every edge the same number of times;
Tree-depth
                       some edges may remain unsubdivided
Orderings

Generalized        a (≤ k )-topological minor of G is any H such that some
coloring
number
                   (≤ k )-subdivision of H is a subgraph of G.
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Admissibility of subdivisions

  Nowhere-
 dense graph   Theorem
 classes and
  algorithms
               If H is a (≤ s − 1)-subdivision of a graph with minimum
  Z. Dvoˇák
        r
               degree d ≥ 3, then adms (H) ≥ d.
Introduction

Subgraph       Proof.
problem

Tree-depth
                   let v be the last vertex of degree at least three in the
Orderings
                   ordering
Generalized
coloring
number
                   the s-backconnectivity of v is at least d
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Subdivisions in high-admissibility graphs

  Nowhere-
 dense graph   Theorem
 classes and
  algorithms
               If adms (G) > (16d)s , then G has a (≤ s − 1)-topological
  Z. Dvoˇák
        r
               minor H such that δ(H) > d.
Introduction

Subgraph       Proof.
problem

Tree-depth
                   Otherwise, average degree of any (≤ s − 1)-topological
Orderings
                   minor is ≤ 2d.
Generalized
coloring
number
                   Greedy algorithm fails, with set M of unchosen vertices:
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Subdivisions in high-admissibility graphs

  Nowhere-
 dense graph
 classes and
  algorithms
                   We have more than (16d)s |M| paths.
        r
  Z. Dvoˇák
                   Almost gives the topological minor, but the paths may
Introduction       overlap.
Subgraph
problem            We need to clean up the paths; consecutively by levels
Tree-depth         (s − 1 times)
Orderings
                   Assuming levels up to i consist of disjoint paths:
Generalized
coloring            1 throw away paths ending with the next step: −4d|M|
number
                    2 if next step of P ends in level j < i of Q, throw away P
Bounded               or Q: /3
expansion
                    3 for each vertex reachable in level i + 1, choose one of
Nowhere-
dense graph           the paths: /4d
classes

Shallow            The resulting graph is too dense.
minors

Closures

Orientations
               Subdivisions in high-admissibility graphs

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Summary

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   Definition
Subgraph
problem        Let s (G) be the largest minimum degree of an
Tree-depth     (≤ s)-topological minor of G.
Orderings

Generalized    We have
coloring
number
                                                                s
Bounded                    s−1 (G)   ≤ adms (G) ≤ (16   s−1 (G)) .
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Definition

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r      Definition
Introduction   A class of graphs C has bounded expansion if there exists a
Subgraph
problem
               function f such that for every s and every G ∈ C,
Tree-depth       s (G) ≤ f (s).
Orderings

Generalized
coloring
               Theorem (Zhu; D.)
number

Bounded
               A class has bounded expansion if and only if it has bounded
expansion      admissibilities.
Nowhere-
dense graph
classes        Recall: bounded admissibilities ⇒ low tree-depth colorings.
Shallow
minors

Closures

Orientations
               Low tree-depth colorings and subdivisions

  Nowhere-
 dense graph
 classes and
               Lemma
  algorithms
               If δ(H) > d, then every subdivision H of H has td(H ) > d.
        r
  Z. Dvoˇák

Introduction
               Proof.
Subgraph
problem        Consider the vertex v of degree greater than d appearing
Tree-depth     deepest in the tree certifying tree-depth of H :
Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Low tree-depth colorings and subdivisions

  Nowhere-
 dense graph   Theorem
 classes and
  algorithms
               If sds (G) has an (s + 2, td ≤ s + 1) coloring by at most c
                                                c
  Z. Dvoˇák
        r
               colors, then δ(G) ≤ 2(s + 1) s+2 .
Introduction

Subgraph       Proof.
problem

Tree-depth
                   For e ∈ E(G), let Se be the set of colors on the
Orderings
                   corresponding path (including endvertices).
Generalized
coloring
number
                   For S ⊆ {1, . . . , c} of size s + 2, let GS be the subgraph
Bounded            with edges {e : Se ⊆ S}.
expansion

Nowhere-
                   For some S, GS has average degree at least
                          c
dense graph
classes
                   δ(G)/ s+2 , and contains G ⊆ GS with
                                       c
Shallow            δ(G ) ≥ δ(G)/ 2    s+2   .
minors

Closures           Observe td(sds (G )) ≤ s + 1 and apply lemma.
Orientations
               Low tree-depth colorings and bounded
               expansion
  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Corollary
Introduction
               If a class of graphs has low tree-depth colorings, then it has
Subgraph
problem        bounded expansion.
Tree-depth

Orderings      I.e., the following are equivalent:
Generalized
coloring
                   bounded expansion
number
                   bounded admissibilities
Bounded
expansion          having low tree-depth colorings
Nowhere-
dense graph        having low tree-width colorings
classes

Shallow
minors

Closures

Orientations
               Classes with bounded expansion

  Nowhere-
 dense graph
 classes and
  algorithms
               Theorem
        r
  Z. Dvoˇák
               Any proper class C of graphs closed on topological minors
Introduction
               has bounded expansion.
Subgraph
problem

Tree-depth     Proof.
Orderings

Generalized        Kk ∈ C for some k
coloring
number
                   if H ∈ C, then δ(H) ≤ O(k 2 ) (Komlós)
Bounded
expansion          closed on topological minors:   s (G)   ≤ O(k 2 ) for every
Nowhere-
dense graph
                   G ∈ C and s ≥ 0.
classes

Shallow
minors

Closures

Orientations
               Classes with bounded expansion

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   Corollary
Subgraph
problem        The following graph classes have bounded expansion:
Tree-depth
                   graphs with bounded maximum degree
Orderings

Generalized
                   proper minor-closed graph classes, e.g.,
coloring
number
                       graphs with bounded tree-width
Bounded
                       planar graphs
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Remark on bounded tree-width

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Bounded tree-width ⇒ bounded expansion ⇒ low
Introduction
               tree-depth colorings, proving
Subgraph
problem

Tree-depth
               Claim
Orderings      There exists a function g such that for every t a p, every
Generalized
coloring
               graph with tree-width at most t has a
number         (p, td ≤ p − 1)-coloring by g(t, p) colors.
Bounded
expansion
               as we promised before.
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Other classes with bounded expansion

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph           graphs drawn in a fixed surface with a bounded number
problem
                   of crossings on each edge
Tree-depth

Orderings
                   created by adding edges in mutual distance ω(1) to
Generalized        graphs in any class with bounded expansion
coloring
number             almost all graphs with linear number of edges
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Generalizations of the subgraph problem

  Nowhere-
 dense graph
 classes and
  algorithms   For all graph classes with bounded expansion (D., Král’,
  Z. Dvoˇák
        r      Thomas):
Introduction
                   testing first-order properties in linear time
Subgraph
                        e.g., having dominating set of size at most k (k fixed):
problem

Tree-depth                  (∃x1 ) . . . (∃xk )(∀y )   y = x1 ∨ . . . ∨ y = xk ∨
Orderings                                              E(y , x1 ) ∨ . . . ∨ E(y , xk ).
Generalized
coloring
number             data structure for graphs with colored vertices and
Bounded
                   edges
expansion
                        linear-time initialization
Nowhere-
dense graph
                        change color of an element in O(1)
classes                 decide first-order query with bounded number of
Shallow                 quantifiers in O(1)
minors

Closures

Orientations
               Nowhere-dense graph classes

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Definition
Introduction

Subgraph       A class of graphs C is nowhere-dense if there exists a
problem
               function f such that for every s, Kf (s) is not an
Tree-depth
               (≤ s)-topological minor of any graph in C.
Orderings

Generalized
coloring
number
                   Equivalently, for every s, the set of (≤ s)-topological
Bounded            minors of graphs in C does not contain all graphs.
expansion
                   Bounded expansion ⇒ nowhere-dense
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Properties of nowhere-dense classes

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   If C is nowhere-dense, then for every ε > 0, integer s and
Subgraph       G ∈ C with n vertices:
problem

Tree-depth           s (G)   = O(nε ), hence
Orderings          adms (G) = O(nε ), hence
Generalized
coloring           G has (s, td ≤ s − 1) coloring by O(nε ) colors, hence
number

Bounded
                   we can test H ⊆ G (for fixed H) in O(n1+ε ).
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Other results

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   Most results for graph classes with bounded expansion also
Subgraph       holds for nowhere-dense graph classes (with O(nε )
problem

Tree-depth
               replacing constants). Exception:
Orderings
               Problem
Generalized
coloring
number
               Are first-order properties FPT on nowhere-dense graph
Bounded
               classes?
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Subgraph problem and nowhere-dense graph
               classes
  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r
               Theorem
Introduction
               Assume that the subgraph problem is not FPT on the class
Subgraph
               of all graphs. If the subgraph problem is FPT on a class of
problem
               graphs C closed on subgraphs, then C is nowhere-dense.
Tree-depth

Orderings

Generalized
               Proof.
coloring
number         For every s ≥ 0,
Bounded
expansion
                                  H ⊆ G ⇔ sds (H) ⊆ sds (G).
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Examples of nowhere-dense classes

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
                   locally bounded expansion, including
Introduction            locally bounded tree-width
Subgraph                locally proper minor closed
problem

Tree-depth

Orderings      Definition
Generalized
coloring
               A class of graphs C has locally bounded expansion if there
number         exists a function f such that for every s, d ≥ 0 and every
Bounded
expansion
               G ∈ C, if H is a subgraph of G of radius at most d, then
Nowhere-         s (H) ≤ f (s, d).
dense graph
classes

Shallow
minors

Closures

Orientations
               Nowhere-dense = bounded expansion

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               The class
Introduction

Subgraph
problem
                C = {G : ∆(G) ≤ log log |V (G)|, girth(G) ≥ log log |V (G)|}
Tree-depth

Orderings          is nowhere-dense: if sds (Kk ) ∈ C, then
Generalized
coloring
                   k − 1 ≤ log log |V (G)| ≤ 3s.
number
                   does not have bounded expansion: unbounded
Bounded
expansion          minimum degree
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Original definition of bounded expansion

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem            Bounded expansion has many different
Tree-depth         characterizations.
Orderings          But we still did not see the one that came the first
Generalized
coloring
                   chronologically.
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Shallow minors

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Definition
Introduction

Subgraph       A depth r minor of G is a graph obtained from a subgraph of
problem
               G by contracting vertex-disjoint subgraphs of radius at most
Tree-depth
               r.
Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Shallow minors

  Nowhere-
 dense graph   Definition
 classes and
  algorithms
               Let r (G) be the greatest average density |E(H)|/|V (H) of
        r
  Z. Dvoˇák
               a depth r minor H of G.
Introduction

Subgraph
               Remark: Greatest Reduced Average Density, hence the
problem        symbol.
Tree-depth

Orderings      Theorem (D.)
Generalized
coloring       For any r ≥ 0 and any graph G,
number
                                                                (r +1)2
Bounded
expansion                     2r (G)   ≤2   r (G)   ≤ 4(4   2r )        .
Nowhere-
dense graph
classes

Shallow
               Proof.
minors
               Idea: split the spanning trees of the shallow minor on
Closures

Orientations
               vertices of big enough degree.
               Shallow minors and bounded expansion

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   Corollary
Subgraph
problem
               A class of graphs C has bounded expansion if and only if
Tree-depth
               there exists a function f such that for every r and every
Orderings
               G ∈ C, r (G) ≤ f (r ).
Generalized
coloring
number         We say that the expansion of C or of the graph G is bounded
Bounded
expansion
               by f .
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Small separators

  Nowhere-
 dense graph   Definition
 classes and
  algorithms
               A set S ⊆ V (G) is a separator if each component of G − S
        r
  Z. Dvoˇák
               has at most 2|V (G)|/3 vertices.
Introduction

Subgraph       Theorem (Plotkin, Rao and Smith)
problem

Tree-depth     If a graph G on n vertices does not contain Kh as depth
Orderings      d log2 n minor, then G has a separator of size at most
Generalized
coloring
               O(n/d + dh2 log n). Can be found in O(|E(G)|n/d).
number

Bounded
expansion
               Corollary
Nowhere-
dense graph
               If there exists c ≥ 0 such that the expansion of G is bounded
classes        by O(r c ), then G has a separator of size (n log n)1−1/(2c+2) .
Shallow
minors
               If the expansion of G is bounded by a subexponential
Closures       function, then G has separator of sublinear size. Tight
Orientations   because of 3-regular expanders.
               Consequences of small separators

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r
               Corollary
Introduction
               If the expansion of G is bounded by a subexponential
Subgraph
problem        function, then G has sublinear tree-width.
Tree-depth

Orderings      Corollary (D., Norine)
Generalized
coloring
number
                                                             log
               For any function f such that lim supr →∞ log log rf (r ) < 1/3,
Bounded        there exists c > 0 such that the number of non-isomorphic
expansion
               graphs G on n vertices with expansion bounded by f is at
Nowhere-
dense graph    most c n .
classes

Shallow
minors

Closures

Orientations
               Blowing up vertices

  Nowhere-
 dense graph   Lemma
 classes and
  algorithms
               Let H be the graph obtained from G by blowing up each
  Z. Dvoˇák
        r
               vertex to a clique of size k . Then s (H) is bounded by
Introduction   a function of s (G) and k .
Subgraph
problem
               Proof.
Tree-depth

Orderings      Let F ⊆ H be an (≤ s)-subdivision of a graph F with
Generalized    δ(F ) = s (H). Each e ∈ E(F ) has a path Pe ⊂ F .
coloring
number             Pe does not contain twins, unless it is an edge
Bounded
expansion          merge twin branchpoints: min. degree ≥ (δ − k + 1)/k
Nowhere-
dense graph
                   remove twins of branchpoints: average degree
classes
                   A ≥ (δ − k + 1)/k − 2(k − 1)
Shallow
minors         Each Pe now conflicts with ≤ (k − 1)s other paths. Choose
Closures       largest subgraph where all paths are independent.
Orientations
               Blowing up vertices

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Contracting a forest

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   Lemma
Subgraph
problem        Let H be the graph obtained from G by contracting a forest.
Tree-depth     Then s (H) is bounded by a function of 3s+5 (G).
Orderings

Generalized
coloring       Proof.
number

Bounded          s (H)   ≤2   s/2   (H) ≤ 2   3 s/2 +1 (G)   ≤ 2f (   3s+5 (G))
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Application

  Nowhere-
 dense graph
 classes and
  algorithms
               Theorem
        r
  Z. Dvoˇák
               For any fixed k ≥ 0, the class of graphs that can be drawn in
Introduction
               plane with at most k crossings on each edge has bounded
Subgraph
problem        expansion.
Tree-depth

Orderings
               Proof.
Generalized
coloring
number
               Put vertices on crossings and subdivide the edges: planar
Bounded
               graph, with bounded expansion. Blow up all vertices to
expansion
               cliques of size two, contract forest: creates crossings.
Nowhere-
dense graph    Suppress vertices of degree two (at most 2k contractions of
classes
               a forest).
Shallow
minors

Closures

Orientations
               Orientations with bounded degree

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r      Claim
Introduction
                 0 (G)≤ d if and only if G has an orientation with indegree
Subgraph
problem
               at most d.
Tree-depth

Orderings
                   compare with: if 0 (G) ≤ d, then G is 2d-degenerate
Generalized
                   (the reverse implication does not hold)
coloring
number
                         equivalently, G has an acyclic orientation with indegree
Bounded
                         ≤ 2d
expansion
                   we will now consider orientations whose acyclic
Nowhere-
dense graph        versions correspond to generalized coloring numbers
classes

Shallow
minors

Closures

Orientations
               Augmentations

  Nowhere-
 dense graph
               Let G be a directed graph. An unordered pair {u, v } is
 classes and
  algorithms       a transitive pair if uw, wv ∈ E(G) for some w ∈ V (G)
  Z. Dvoˇák
        r          a fraternal pair if uw, vw ∈ E(G) for some w ∈ V (G)
Introduction

Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number
               Definition
Bounded
expansion      A directed graph H is an augmentation of G if the edge set
Nowhere-
dense graph
               of the underlying undirected graph of H consists of the
classes
               edges of G and of all transitive and fraternal pairs in G.
Shallow
minors
               I.e., add the transitive and fraternal pairs as edges and give
Closures

Orientations
               them arbitrary orientations.
               Density of augmentations

  Nowhere-
 dense graph   Theorem
 classes and
  algorithms
               If H is the underlying undirected graph of an augmentation
  Z. Dvoˇák
        r
               of G and d is the maximum indegree of G, then s (H) is
Introduction   bounded by a function of d and 3s+5 (G).
Subgraph
problem
               Proof.
Tree-depth

Orderings      H is a subgraph of graph obtained by replacing each vertex
Generalized    by d + 1 vertices and contracting a star forest:
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Steady augmentations

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction   Theorem
Subgraph
problem
               If H is the underlying undirected graph of an augmentation
Tree-depth     of G and d is the maximum indegree of G, then s (H) is
Orderings      bounded by a function of d and 3s+5 (G).
Generalized
coloring
number
               In particular, there exists an augmentation of G with
Bounded
               maximum indegree bounded by a function of d and         5 (G).
expansion
               We call such an augmentation steady.
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Short paths via augmentations

  Nowhere-
 dense graph   Lemma
 classes and
  algorithms
               Let s ≥ 2, let G be a graph, G0 its orientation and G0 , G1 ,
  Z. Dvoˇák
        r
               G2 , . . . , Gs a sequence of augmentations. If the distance
Introduction   between u and v in G is at most (3/2)s−1 + 2, then u and v
Subgraph       are either adjacent or have a common in-neighbor in Gs .
problem

Tree-depth

Orderings
               Proof.
Generalized    In each augmentation, the path of length t gives rise to a
coloring
number         path of length ≤ 2/3t + 2/3:
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Oracle for short paths

  Nowhere-
 dense graph
 classes and
  algorithms

  Z. Dvoˇák
        r      Fix s ≥ 2. Find an orientation G0 of G with bounded
Introduction
               indegree and compute augmentations G1 , . . . , Gs .
Subgraph           for each edge, remember the length of the shortest
problem
                   corresponding path
Tree-depth

Orderings          plus one of the ways how it was created
Generalized        for bounded expansion classes, if steady
coloring
number             augmentations are used:
Bounded                 linear-time preprocessing
expansion
                        by Lemma, O(1) queries for paths of length at most
Nowhere-
dense graph             (3/2)s−1 + 2 (every vertex has only O(1) in-neighbors).
classes

Shallow
minors

Closures

Orientations
               Dynamic version

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction
                  idea: maintain the augmentations when edges are
Subgraph
problem           added or removed
Tree-depth        problem: adding edge can create unbounded number
Orderings
                  of edges due to transitive pairs
Generalized
coloring          solution: only add edges for fraternal pairs (fraternal
number

Bounded
                  augmentation).
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Short paths via fraternal augmentations

  Nowhere-
 dense graph
 classes and   Lemma
  algorithms

  Z. Dvoˇák
        r
               Let s ≥ 0, let G be a graph, G0 its orientation and G0 , G1 ,
               G2 , . . . , Gs a sequence of fraternal augmentations. If the
Introduction
               distance between u and v in G is at most s + 1, then there
Subgraph
problem        exists a path P = w1 w2 . . . wt between u = w1 and v = wt in
Tree-depth     Gs of length at most s + 1, and an index c ≤ t such that the
Orderings      edges w1 w2 , w2 w3 , . . . , wc−1 wc are oriented towards u and
Generalized
coloring
               the rest of the edges is oriented towards v .
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors         To find the path, search in-neighbors up to distance s + 1
Closures
               (O(1) if steady augmentations are used)
Orientations
               Maintaining the orientation

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Theorem (Brodal and Fagerberg)
Introduction

Subgraph       For any d > 0 there exists D so that an orientation of a
problem
               d-degenerate graph on n vertices with maximum indegree
Tree-depth

Orderings
               D can be maintained within the following time bounds:
Generalized
coloring
                   an edge can be added in O(log n) (amortized)
number
                   an edge can be removed in O(1)
Bounded
expansion
               Each vertex stores a list of in- and out-neighbors.
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Maintaining the augmentations

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

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

Tree-depth
                   each reorientation adds or removes O(1) edges in G1
Orderings          O(log2 n) reorientations in G1 , . . .
Generalized
coloring       Gives O(logs n) update time for maintaining paths of length
number
               at most s.
Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Low tree-depth colorings via augmentations

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák
               Theorem
Introduction
               Let s ≥ 1, let G be a graph, G0 its orientation and G0 , G1 ,
Subgraph
problem        G2 , . . . , Gp a sequence of augmentations, where
Tree-depth     p = 3(s + 1)2 . Let H be the underlying undirected graph of
Orderings      Gp . Then any proper coloring of H gives an
Generalized
coloring
               (s, td ≤ s − 1)-coloring of G.
number

Bounded        For bounded expansion classes, linear-time and the number
expansion
               of colors is bounded, if steady augmentations are used.
Nowhere-
dense graph    Proof is lengthy and technical.
classes

Shallow
minors

Closures

Orientations
               Low tree-depth colorings via augmentations

  Nowhere-
 dense graph   Theorem
 classes and
  algorithms
               Let s ≥ 1, let G be a graph, G0 its orientation and G0 , G1 ,
  Z. Dvoˇák
        r
               G2 , . . . , Gp a sequence of augmentations, where p = s+1 .
                                                                         2
                                                          2s
Introduction
               Any proper coloring of Gp is an s, td ≤    2    -coloring of G.
Subgraph
problem

Tree-depth     Proof.
Orderings

Generalized        Show that no path on 2s vertices uses ≤ s colors, since
coloring
number
                   the subgraph induced by the path contains Ks+1 .
Bounded                    s
expansion          After   2   augmentations we have two disjoint Ks ;
Nowhere-
dense graph        have directed Hamiltonian paths, starts v1 and v2 ,
classes

Shallow
                   v1 and v2 adjacent or a common in-neighbor.
minors
                   Say v1 has an in-neighbor w outside of its clique,
Closures

Orientations
                   next s augmentations to add w to the clique.
               Low tree-depth colorings via augmentations

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction

Subgraph
problem

Tree-depth

Orderings

Generalized
coloring
number

Bounded
expansion

Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations
               Summary

  Nowhere-
 dense graph
 classes and
  algorithms

        r
  Z. Dvoˇák

Introduction      Bounded expansion and nowhere-dense serve as good
Subgraph          formalization of “structurally sparse” graphs.
problem

Tree-depth
                  Many natural graph classes have these properties.
Orderings         Problems expressible in first-order logic can be solved
Generalized
coloring
                  efficiently for them.
number
                  Many other results for special classes of sparse graphs
Bounded
expansion         generalize to this setting.
Nowhere-
dense graph
classes

Shallow
minors

Closures

Orientations

								
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