VIEWS: 33 PAGES: 83 POSTED ON: 8/11/2011
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