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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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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)
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Subgraph problem
Problem
Input: graphs H and G.Question: is H a subgraph of G?
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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 .
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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?
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Tree-depth versus tree-width
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Tree-depth versus tree-width
Corollary (of the Corollary)
A class of graphs has low tree-width colorings if and only if ithas low tree-depth colorings.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Arrangeability
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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:
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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?
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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:
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Admissibility
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Admissibility
Observation: greedy algorithm correctly determinesadms(G)
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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?
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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:
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Subdivisions in high-admissibility graphs
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Subdivisions in high-admissibility graphs
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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 ′:
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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)
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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?
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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 .
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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 .
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Blowing up vertices
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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))
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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:
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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).
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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)
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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, 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.
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
Shallowminors
Closures
Orientations
Conclusions
Low tree-depth colorings via augmentations
Nowhere-dense graphclasses andalgorithms
Z. Dvorák
Introduction
Subgraphproblem
Tree-depth
Orderings
Generalizedcoloringnumber
Boundedexpansion
Nowhere-dense graphclasses
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.