Nowhere dense graph classes and algorithmic applications. A tutorial at Highlights of Logic, Games and Automata 2019
aa r X i v : . [ c s . D M ] S e p Nowhere dense graph classes andalgorithmic applications
A tutorial at Highlights of Logic, Games and Automata 2019
Sebastian SiebertzUniversity of Bremen [email protected]
Abstract.
The notion of nowhere dense graph classes was introduced by Nešetřiland Ossona de Mendez and provides a robust concept of uniform sparseness of graphclasses. Nowhere dense classes generalize many familiar classes of sparse graphs suchas classes that exclude a fixed graph as a minor or topological minor. They admitseveral seemingly unrelated natural characterizations that lead to strong algorith-mic applications. In particular, the model-checking problem for first-order logic isfixed-parameter tractable over these classes. These notes, prepared for a tutorialat Highlights of Logic, Games and Automata 2019, are a brief introduction to thetheory of nowhere denseness, driven by algorithmic applications.
The notion of excluded minors is celebrated as one of the most successful notions in contemporarygraph theory and has an immense influence on algorithmic graph theory. At its heart lies thestructure theorem that states that every graph G that excludes a fixed graph H as a minorcan be decomposed in a treelike way into parts that can be almost topologically embedded ona surface that H does not embed on [35]. Surprisingly, the theory of bounded expansion and nowhere dense graph classes , i.e. the theory of bounded depth minors , which is much simpler andyet deals with much more general graph classes, is much less known. The notions of boundedexpansion and nowhere denseness were introduced by Nešetřil and Ossona de Mendez [26, 27]and provide a robust concept of uniform sparseness of graph classes. Classes with boundedexpansion and nowhere dense classes generalize many familiar classes of sparse graphs, such asclasses that exclude a fixed graph as a minor or topological minor. They admit several naturalcharacterizations that lead to strong algorithmic applications. In particular, the model-checkingproblem for first-order logic is fixed-parameter tractable over these classes [10, 19].In this short exposition I would like to give a very accessible introduction to the theory ofbounded expansion and nowhere denseness. The presentation is driven by the application ofsolving the first-order model-checking problem based on Gaifman’s locality theorem. Therefore,I focus on the aspect of appropriately localizing well known width measures from graph theory.The original definitions of bounded expansion and nowhere dense classes are given by imposingrestrictions on the bounded depth minors that can be found in graphs from the class. Analgorithmically very useful equivalent definition of nowhere dense classes is given in terms of uniform quasi-wideness , which is often considered as one of the more cumbersome parts ofthe theory. I will present this concept as a local version of treewidth and as a local versionof treedepth, and hope to convince the reader of the beauty of the concept. Finally, a thirdcharacterization is provided in terms of weak reachability numbers, which can again be seen asa local version of treedepth. 1 First-order model-checking
First-order logic can express many interesting algorithmic properties of graphs such as the exis-tence of an independent set of size at least k , the existence of a (connected) dominating set ofsize at most k , and many more. For example, the formula ϕ k dom := ∃ x . . . ∃ x k ∀ y (cid:0) _ ≤ i ≤ k ( y = x i ∨ E ( y, x i ) (cid:1) is true in a graph G if and only if G has a dominating set of size at most k . The model-checkingproblem for first-order logic is the problem to test for an input structure A and input formula ϕ whether ϕ holds in A , in symbols A | = ϕ .The model-checking problem for first-order logic on an input structure A of order n is decidablein time n O ( q ) , where q is the quantifier rank of the formula ϕ . Phrased in terms of parameterizedcomplexity, the problem belongs to the complexity class XP of slicewise polynomial problems.It is expected that in general this running time cannot be avoided, e.g. testing whether a graphcontains a clique with k vertices, which requires exactly k quantifiers, cannot be done in time n o ( k ) unless the exponential time hypothesis fails [4].This has led to the investigation of structural properties of the input structures, especiallyof graphs, that allow for more efficient model-checking. In particular, we search for (the mostgeneral) graph classes on which the problem can be solved in time f ( | ϕ | ) · n c for some computablefunction f and constant c , i.e. for classes where the problem is fixed-parameter tractable (param-eterized by the formula length). As tractability for the model-checking problem of a logic impliestractability not only for individual problems but for whole classes of problems, a tractability re-sult for a model-checking problem is often referred to as an algorithmic meta theorem . There is along line of meta theorems for first-order logic on sparse structures [5, 10, 13, 14, 20, 36], culminat-ing in the result that the model-checking problem for first-order logic is fixed-parameter tractableon every nowhere dense class of graphs [19]. It was shown earlier that on every subgraph-closedgraph class that is not nowhere dense the problem is as hard as on all graphs [10, 20], hence,the classification of tractability for the first-order model-checking problem on subgraph-closedclasses is essentially complete.The key property of first-order logic that is exploited for efficient model-checking is locality .Gaifman’s Locality Theorem states that every first-order formula ϕ (¯ x ) is equivalent to a Booleancombination of1. local formulas ψ ( r ) (¯ x ) and2. basic local formulas ∃ x . . . ∃ x k (cid:0) V i = j dist( x i , x j ) > r ∧ χ ( r ) ( x i ) (cid:1) .Here, the notation ψ ( r ) (¯ x ) means that for every graph G and every tuple ¯ v ∈ V ( G ) | ¯ x | wehave G | = ψ ( r ) (¯ v ) if and only if G [ N r (¯ v )] | = ψ ( r ) (¯ v ) , where G [ N r (¯ v )] denotes the subgraphof G induced by the r -neighborhood N r (¯ v ) of ¯ v . This property is syntactically ensured in ψ ( r ) by relativizing all quantifiers to distance at most r from one of the the free variables. Thenumbers r and k in the formulas above depend only on the formula ϕ , and furthermore, theGaifman normal form of any formula ϕ is computable from ϕ .This translates the model-checking problem to the following algorithmic problem. To decidefor a graph G and tuple ¯ v ∈ V ( G ) | ¯ x | whether G | = ϕ (¯ v ) ,1. decide whether ¯ v has the local properties described by ψ ( r ) (¯ x ) ;2. decide for each v ∈ V ( G ) whether G | = χ ( r ) ( v ) ;3. solve each generalized independent set problem described by the basic local formulas ∃ x . . . ∃ x k (cid:0) V i = j dist( x i , x j ) > r ∧ χ ( r ) ( x i ) (cid:1) , and finally4. evaluate the Boolean combination of these statements that is equivalent to ϕ .2 Bounded depth minors, bounded expansion and nowheredenseness
By Gaifman’s theorem, we expect to have efficient model-checking algorithms on graph classesthat locally have nice properties. With this motivation in mind we can try to find appropriatelocal versions of width measures that we know how to handle well. This approach was followede.g. in [5, 14] where it was simply required that the r -neighborhoods in graphs from the classhave good properties, e.g. they have bounded treewidth, or exclude a minor. For example, wesay that a class C has locally bounded treewidth if for every r ∈ N there exists a number t = t ( r ) such that for every G ∈ C and every v ∈ V ( G ) the treewidth of G [ N r ( v )] is bounded by t .Similarly, we say that a class C locally excludes a minor if for every r ∈ N there exists a number m = m ( r ) such that for every G ∈ C and every v ∈ V ( G ) the graph G [ N r ( v )] excludes thecomplete graph K m on m vertices as a minor (the concepts of treewidth and minors are definedformally below). Note however, that this approach of defining locally well behaved classes is notvery robust. For example, if we add to every graph G ∈ C an apex vertex, i.e. a vertex that isconnected with every other vertex of G , then the resulting class has locally bounded treewidthif and only if the class C has bounded treewidth. On the other, it is very easy to algorithmicallyhandle the apex vertices and we are looking for more robust locality notions.The following notion of bounded depth minors is the fundamental notion in the theory ofbounded expansion and nowhere denseness [26, 27]. Definition 1.
A graph H is a minor of G , written H G , if there is a map φ that assigns toevery vertex v ∈ V ( H ) a connected subgraph φ ( v ) ⊆ G of G and to every edge e ∈ E ( H ) anedge φ ( e ) ∈ E ( G ) such that1. if u, v ∈ V ( H ) with u = v , then V ( φ ( v )) ∩ V ( φ ( u )) = ∅ and2. if e = uv ∈ E ( H ) , then φ ( e ) = u ′ v ′ ∈ E ( G ) for some u ′ ∈ V ( φ ( u )) and v ′ ∈ V ( φ ( v )) .The set φ ( v ) for a vertex v ∈ V ( H ) is called the branch set or model of v in G . The map φ is called the minor model of H in G . The depth of a minor model is the maximal radius of itsbranch sets. For r ∈ N , the graph H is a depth- r minor of G , written H r G , if there is aminor model φ of H in G of depth at most r .Now, bounded expansion and nowhere dense classes are defined by imposing restrictions onthe structure of bounded depth minors. Definition 2.
A class C of graphs has bounded expansion if for every r ∈ N there exists number d = d ( r ) such that the edge density | E ( H ) | / | V ( H ) | of every H r G for G ∈ C is bounded by d . Definition 3.
A class C of graphs is nowhere dense if for every r ∈ N there exists a number m = m ( r ) such that we have K m r G for all G ∈ C . Example 4.
1. Every class C that excludes a fixed graph H as a minor has bounded expansion. Forsuch classes there exists an absolute constant c such that for all r ∈ N the edge densityof depth- r minors of graphs in C is bounded by c . Special cases are classes of boundedtreewidth, the class of planar graphs, and every class of graphs that can be drawn with abounded number of crossings, see [31], and every class of graphs that embeds into a fixedsurface.2. Every class C that excludes a fixed graph H as a topological minor has bounded expansion.Every class that excludes H as a minor also excludes H as a topological minor. Furtherspecial cases are classes of bounded degree and classes of graphs that can be drawn witha linear number of crossings, see [31]. 3. Every class of graphs that can be drawn with a bounded number of crossings per edge hasbounded expansion [31].4. Every class of graphs with bounded queue-number, bounded stack-number or boundednon-repetitive chromatic number has bounded expansion [31].5. The class of Erdös-Rényi random graphs with constant average degree d/n , G ( n, d/n ) , hasasymptotically almost surely bounded expansion [31].6. Every bounded expansion class is nowhere dense.7. The class of graphs with girth greater than maximum degree is nowhere dense (and haslocally bounded treewidth) and does not have bounded expansion [28].Nowhere dense classes can also be defined in terms of subdivisions or topological minors . Thisfact is very useful for proving algorithmic lower bounds for classes that are not nowhere dense.For r ∈ N , a graph H is an r -subdivision of a graph G if H is obtained from G by replacingevery edge by a path of length r + 1 (containing r inner vertices). Lemma 5.
Let C be a class that is not nowhere dense and that is closed under taking subgraphs.Then there exists r ∈ N such that C contains an r -subdivision of every graph H . Using the lemma it is for example not difficult to show that the first-order model-checkingproblem on every class that is not nowhere dense and closed under taking subgraphs is as hardas on the class of all graphs.Finally, we note that nowhere dense classes are sparse.
Theorem 6 ([7, 27]).
A class C of graphs is nowhere dense if and only if for all real ǫ > and all r ∈ N there exists an integer n such that all n -vertex graphs H r G for G ∈ C with n > n have edge density at most n ǫ . At this point the notions of bounded expansion and nowhere denseness are established asabstract concepts. Observe that we have achieved the desired robustness of the concepts undersmall changes, such as adding apex vertices to the graphs of a class C . On the other handobserve that we cannot expect to find a structure theorem as for classes that exclude a fixedminor H . For example the class of graphs that contains the n -subdivision of every n -vertexgraph G has bounded expansion and we cannot find a global decomposition for the graphs fromthis class. This example also shows the limitations for algorithmic applications. We will e.g. notbe able to solve global connectivity problems more efficiently than on general graph classes. Wewill now move to the tools that can be used to handle bounded expansion and nowhere densegraph classes. The separator width of a graph G is defined as the minimum number k such that for every A ⊆ V ( G ) there exists a set S of order at most k such that for every component C of G − S wehave | V ( C ) ∩ A | ≤ | A | / . A class of graphs has bounded treewidth if and only if it has boundedseparator width. In fact, the main algorithmic applications of graphs with bounded treewidthfollow from the property that these graphs admit small balanced separators. Following our goalof finding an appropriate localization of this property we give the following definition. Definition 7.
A class C of graphs admits balanced neighborhood separators if for every r ∈ N and every real ǫ > there exists a number s = s ( r, ǫ ) such that the following holds. For everygraph G ∈ C and every subset A ⊆ V ( G ) there exists a set S ⊆ V ( G ) of order at most s suchthat the | N r ( v ) G − S ∩ A | ≤ ǫ | A | for all v ∈ V ( G ) \ S .4 heorem 8 ([29]). A class C of graphs is nowhere dense if and only if C admits balancedneighborhood separators. The proof uses the above mentioned characterization of nowhere dense classes in terms ofuniform quasi-wideness that we define next. If G is a graph and A ⊆ V ( G ) , then A is distance- r independent if the vertices of A have pairwise distance greater than r in G . Definition 9.
A graph class C is uniformly quasi-wide if for all r, m ∈ N there exist numbers s = s ( r ) and N = N ( r, m ) such that the following holds. For every graph G ∈ C and every set A ⊆ V ( G ) : if | A | ≥ N , then there exists S ⊆ V ( G ) with | S | ≤ s and B ⊆ A \ S with | B | ≥ m such that B is distance- r independent in G − S . Theorem 10 ([27]).
A class C of graphs is nowhere dense if and only if C is uniformly quasi-wide. We are not going to prove Theorem 10 as the proof is quite technical. However, to get familiarwith the concept of uniform quasi-wideness it is instructive to prove Theorem 8.
Proof. (of Theorem 8)
Let C be nowhere dense and let r ∈ N and ǫ > . According toTheorem 10, C is uniformly quasi-wide. Hence, for r ′ = 4 r there exists s = s ( r ′ ) and for m = ⌊ /ǫ ⌋ + s + 1 there exists N = N ( r ′ , m ) such that for every graph G ∈ C and every X ⊆ V ( G ) : if | X | ≥ N , then there exists Y ⊆ V ( G ) with | Y | ≤ s and X ′ ⊆ X \ Y with | X ′ | ≥ m such that X ′ is distance- r independent in G − Y .Let G ∈ C and A ⊆ V ( G ) . We aim to prove that there exists S ⊆ V ( G ) with | S | ≤ N suchthat | N G − Sr ( v ) ∩ A | ≤ ǫ | A | for all v ∈ V ( G ) \ S .Let X ⊆ V ( G ) be any set such that for all v ∈ V ( G ) \ X we have | N r ( v ) G − X ∩ A | ≤ ǫ | A | . Weshow that if | X | > N , then there exists Z ⊆ V ( G ) with | Z | < | X | such that for all v ∈ V ( G ) \ Z we have | N G − Zr ( v ) ∩ A | ≤ ǫ | A | . The claim follows by repeating the argument until | Z | ≤ N andthen setting S = Z .If | X | > N , then there exist sets Y ⊆ V ( G ) with | Y | ≤ s and X ′ ⊆ X \ Y with | X ′ | ≥⌊ /ǫ ⌋ + s + 1 that is distance- r independent in G − Y . In X ′ there are at most ⌊ /ǫ ⌋ vertices v with | N G − Y r ( v ) ∩ A | ≥ ǫ | A | , as these neighborhoods are disjoint. Hence, since | X ′ | ≥ ⌊ /ǫ ⌋ + s +1 ,there is a subset X ′′ ⊆ X ′ with | X ′′ | > s and such that | N G − Y r ( v ) ∩ A | ≤ ǫ | A | for all v ∈ X ′′ .Let Z = ( X \ X ′′ ) ∪ Y .We claim that every v ∈ V ( G ) \ Z satisfies | N r ( v ) G − Z ∩ A | ≤ ǫ | A | . To see this, let v ∈ V ( G ) \ Z and assume N G − Zr ( v ) ∩ A = N G − Xr ( v ) ∩ A (we have to consider only such elements, as byassumption | N G − Xr ( v ) ∩ A | ≤ ǫ | A | for all v ∈ V ( G ) \ X ). This implies that there is x ∈ X ′′ such that dist G − Z ( v, x ) < r . This implies N G − Zr ( v ) ⊆ N G − Z r ( x ) . By construction we have | N G − Z r ( x ) ∩ A | ≤ ǫ | A | , as claimed.Vice versa, assume C is not nowhere dense. We show that C does not admit balancedneighborhood covers. Let s : N × R → N be an arbitrary function. According to Lemma 5, thereis r ∈ N such that C contains an r -subdivision of every graph H . Let n := 2 s (2 r, / . Let G ∈ C be such that an r -subdivision of K n is a subgraph of G . Let A be a set of vertices of G thatcontains the vertices of this subdivision. Let S be any set of size at most s (2 r, / . Then thegraph G [ A \ S ] contains a vertex whose r -neighborhood has order at least | A |− n − (cid:0) n/ (cid:1) r > | A | / .Hence, s is not a function for choosing s = s ( r, ǫ ) for balanced neighborhood separators. As s was chosen arbitrary, this proves the claim. (cid:3) The proof of Theorem 8 can be made algorithmic: we algorithmically iterate the exchangeargument of the proof until we arrive at a set of order at most N . In each step we need tocompute a set Y (the set S in the definition of uniform quasi-wideness). This can be done inpolynomial time, see [22, 34]. The work [34] gives also the best known bounds for the function N in the definition of uniform quasi-wideness. 5he algorithmic applications lie at hand. We can recursively decompose local neighborhoodsinto smaller and smaller pieces such that the recursion stops after log n steps. In the next sectionwe will see that we can do even better and get a recursion tree of depth depending only on r . Graph classes whose members admit tree decompositions of bounded width and bounded depthare called classes with bounded treedepth . The notion of treedepth was introduced by Nešetřiland Ossona de Mendez in [25], equivalent notions were studied before under different names.We refer to [28] for a discussion on the various equivalent parameters.A rooted tree T is an acyclic connected graph with one designated root vertex. This imposesthe standard ancestor/descendant relation in T : a node v is a descendant of all the nodes thatappear on the unique path leading from v to the root. A rooted forest F is a disjoint union ofrooted trees. We write u ≤ F v if u is an ancestor of v in F . The relation ≤ F is a partial orderon the nodes of F with the roots being the ≤ F -minimal elements. The depth of a vertex v in arooted forest F is the number of vertices on the path from v to the root (of the tree to which v belongs). The depth of F is the maximum depth of the vertices of F . Definition 11.
Let G be a graph. The treedepth td( G ) of G is the minimum depth of a rootedforest F on the same vertex set as G such that whenever uv ∈ E ( G ) , then u ≤ F v or v ≤ F u .We can equivalently define treedepth by the following elimination game. Let ℓ ∈ N . The ℓ -round treedepth game on a graph G is played by two players, connector and splitter , as follows.We let G := G . In round i + 1 of the game, connector chooses a component C i +1 of G i .Then splitter picks a vertex w i +1 ∈ V ( C i +1 ) . We let G i +1 := C i +1 − { w i +1 } . Splitter winsif G i +1 = ∅ . Otherwise the game continues at G i +1 . If splitter has not won after ℓ rounds, thenconnector wins.A strategy for splitter is a function σ that maps every partial play ( C , w , . . . , C s , w s ) , withassociated sequence G , . . . , G s of graphs, and the next move C s +1 of connector, to a vertex w s +1 ∈ V ( C s +1 ) that is the next move of splitter. A strategy σ is a winning strategy for splitterif splitter wins every play in which she follows the strategy f . We say that splitter wins thesimple ℓ -round radius- r splitter game on G if she has a winning strategy. Lemma 12 (Folklore).
A graph G has treedepth ℓ if and only if splitter wins the ℓ -roundtreedepth game on G . We now consider the following change of the rules of the game that is motivated by our goalto find an appropriate localization of treedepth. The game gets an additional parameter r forthe radius. Instead of picking in round i + 1 of the game a component C i +1 of the currentlyconsidered graph G i , connector picks a subgraph C i +1 of radius at most r in G i . Formally, weconsider the following game.Let ℓ, r ∈ N . The simple ℓ -round radius- r splitter game on a graph G is played by two players, connector and splitter , as follows. We let G := G . In round i + 1 of the game, connector choosesa subgraph C i +1 of G i of radius at most r . Then splitter picks a vertex w i +1 ∈ V ( C i +1 ) . Welet G i +1 := C i +1 − { w i +1 } . Splitter wins if G i +1 = ∅ . Otherwise the game continues at G i +1 . Ifsplitter has not won after ℓ rounds, then connector wins. Strategies are defined as above. Theorem 13 ([19]).
A class C of graphs is nowhere dense if and only if for every r ∈ N thereexists a number ℓ = ℓ ( r ) such that splitter wins the simple ℓ -round radius- r splitter game onevery graph G ∈ C . Proof.
For convenience we allow splitter in every round i to delete not a single vertex w i buta set W i of m ( r ) vertices for any fixed function m . Obviously this does not give him additionalpower, as he can simulate the deletion of m vertices in m rounds of the game.6et r ∈ N . As C is nowhere dense, it is also uniformly quasi-wide. Let s = s ( r ) and N = N ( r, s + 2) be the numbers satisfying the properties of Definition 7. Let ℓ := N and m ( r ) := ℓ · ( r + 1) . Note that both ℓ and m only depend on C and r . We claim that forany G ∈ C , splitter wins the ℓ -round radius- r splitter game in which splitter is allowed todelete m ( r ) vertices in each round.Let G ∈ C be a graph. In the game on G , splitter uses the following strategy. In the firstround, if connector chooses a subgraph C of G = G of radius at most r , say rooted at a vertex v ∈ V ( C ) , i.e. V ( C ) ⊆ N r ( v ) , then splitter chooses the set W := { v } . Now let i > andsuppose that v , . . . , v i , G , . . . , G i , W , . . . , W i have already been defined. Suppose connectorchooses a subgraph C i +1 of G i , say rooted at v i +1 ∈ V ( G i ) . We define W i +1 as follows. Foreach ≤ j ≤ i , choose a path P j,i +1 in C j of length at most r connecting v j and v i +1 . Such a pathmust exist as v i +1 ∈ V ( C i ) ⊆ V ( C j ) ⊆ N G j − r ( v j ) . We let W i +1 := S ≤ j ≤ i V ( P j,i +1 ) ∩ V ( C i +1 ) .Note that | W i +1 | ≤ i · ( r + 1) (the paths have length at most r and hence consist of r + 1 vertices).It remains to be shown that the length of any such play is bounded by ℓ .Assume towards a contradiction that connector can survive on G for ℓ ′ = ℓ + 1 rounds.Let ( v , . . . , v ℓ ′ , G , . . . , G ℓ ′ , W , . . . , W ℓ ′ ) be the play. As ℓ ′ > N ( r, s +2) , for W := { v , . . . , v ℓ ′ } there is a set S ⊆ V ( G ) with | S | ≤ s , such that W contains an r -independent set I of size t :=2 s + 2 in G − S . Without loss of generality assume that I = { v , . . . , v ℓ ′ } .We now consider the pairs ( v j − , v j ) for ≤ j ≤ s + 1 . By construction, P j := P j − , j isa path of length at most r from v j − to v j in G j − . Any path P j must necessarily containa vertex s j ∈ S , as otherwise the path would exist in G − S , contradicting the fact that I is r -independent in G − S . We claim that for i = j , s i = s j , but this is not possible, as thereare at most s vertices in S . To prove the claim, assume i > j . Then V ( P j ) ∩ V ( G j − ) ⊆ W j ,thus V ( P j ) ∩ V ( G j ) = ∅ , and V ( P i ) ⊆ V ( G i − ) ⊆ V ( G j ) . Thus V ( P i ) ∩ V ( P j ) = ∅ for i = j . (cid:3) It is easy to see that the strategy of splitter is efficiently computable, as it amounts to com-puting breadth-first searches in the subgraphs arising in the game. The splitter game allowsto recursively decompose local neighborhoods such that the recursion tree has bounded depth.This can be used for example to solve the generalized distance- r independent set problem thatarises as a problem in the model-checking algorithm.For the general model-checking problem we still have to deal with two combinatorial problems.The first problem is the following. In a naive approach we would translate an input formula ϕ intoGaifman normal form and for each of the local formulas χ ( r ) ( x ) and for each vertex v ∈ V ( G ) tryto evaluate whether G | = χ ( r ) ( v ) . This is equivalent to evaluating whether G [ N r ( v )] | = χ ( r ) ( v ) .We would treat the r -neighborhood of each vertex v as the first move of connector in the splittergame and delete splitter’s answer from G [ N r ( v )] . By marking the neighbors of all deleted verticeswe can translate the formula χ to an equivalent formula χ ′ over an extended vocabulary. We thentranslate χ ′ again into Gaifman normal form and recurse. The first problem of this approach isthat when translating ϕ into Gaifman normal form, we introduce new quantifiers to syntacticallylocalize the formula χ ( r ) . This leads to a higher locality radius r ′ when translating the formula χ ′ again into Gaifman normal form, and so on. Hence, we cannot play the splitter game with theconstant radius r in this naive approach. The second problem is that even if we fixed the firstproblem the resulting algorithm would have a worst-case running time of n O ( ℓ ( r )) , as we create arecursion tree with worst-case branching degree n and depth ℓ ( r ) . This is no improvement overthe simple algorithm running in time n O ( | ϕ | ) .The first problem is handled as follows. We know that the new quantifiers that are used in χ are only used to localize the formula, that is, to express distance constraints. We can thereforeenrich first-order logic by atoms to express distances, so that we do not waste quantifiers forlocalization. We have to be careful though, as these new quantifiers bring additional powerto our formulas. The clue is to define a new rank function (instead of quantifier rank) thatlimits the use of distance atoms in the scope of quantifiers. Intuitively, the more quantifiers7re available in a subformula (of original first-order logic), the larger distances the formula canexpress. By carefully choosing the rank function we get a modified version of Gaifman’s localitytheorem such that the rank remains stable under localization.The second problem is handled as follows. We cannot afford a branching degree n in the recur-sion, but instead we must group closeby vertices that share many vertices in their r -neighborhoodsin clusters. This concept is captured by the notion of neighborhood covers that is explained next. The existence of sparse neighborhood covers for nowhere dense graph classes is derived from asecond characterization of treedepth via elimination orderings . The appropriate local version ofthis measures leads to the definition of weak coloring numbers . Let me define sparse neighborhoodcovers first.
Definition 14.
For r ∈ N , an r -neighborhood cover X of a graph G is a set of connectedsubgraphs of G called clusters , such that for every vertex v ∈ V ( G ) there is some X ∈ X with N r ( v ) ⊆ V ( X ) . The radius rad( X ) of a cover X is the maximum radius of any of itsclusters. The degree d X ( v ) of v in X is the number of clusters that contain v . A class C admitssparse neighborhood covers if there exists c ∈ N and for all r ∈ N and all real ǫ > a number d = d ( r, ǫ ) such that every n -vertex graph G ∈ C admits an r -neighborhood cover of radius atmost c · r and degree at most d · n ǫ . Theorem 15 ([19, 18]).
A class C is nowhere dense if and only if the class C ⊆ = { H ⊆ G : G ∈ C } admits sparse neighborhood covers. The proof of the theorem is based on a characterization of nowhere dense classes in terms ofweak coloring numbers, which can be seen as another local version of treedepth. An order ofthe vertex set V ( G ) = { v , . . . , v n } of an n -vertex graph G is a permutation π = ( v , . . . , v n ) .We say that v i is smaller than v j and write v i < π v j if i < j . We write Π( G ) for the set of allorders of V ( G ) . The coloring number col( G ) of a graph G is the minimum integer k such thatthere exists a linear order π of the vertices of G , such that every vertex v has back-degree atmost k − , i.e., at most k − neighbors u with u < π v . The coloring number of G minus one isequal to the degeneracy of G , which is the minimum integer ℓ such that every subgraph H ⊆ G has a vertex of degree at most ℓ . Definition 16.
Let G be a graph and let π be an order of V ( G ) . We say that a vertex u ∈ V ( G ) is weakly reachable with respect to π from a vertex v ∈ V ( G ) if u ≤ π v and there exists a path P between u and v with w > π u for all internal vertices w ∈ V ( P ) . We write WReach[
G, π, v ] forthe set of vertices that are weakly reachable from v . The depth of π on G is the maximum overall vertices v of G of | WReach[
G, π, v ] | . Lemma 17 (see e.g. [28], Lemma 6.5).
Let G be a graph. The treedepth of G is equal to theminimum depth over all orders π of V ( G ) . We can naturally define a local version of weak reachability.
Definition 18.
Let G be a graph and r ∈ N . Let π be a linear order of V ( G ) . We say that avertex u ∈ V ( G ) is weakly r -reachable with respect to π from a vertex v ∈ V ( G ) if u ≤ π v andthere exists a path P between u and v of length at most r with w > π u for all internal vertices w ∈ V ( P ) . The set of vertices weakly r -reachable by v with respect to the order π is denoted WReach r [ G, π, v ] . We define wcol r ( G, π ) := max v ∈ V ( G ) | WReach r [ G, π, v ] | , weak r -coloring number wcol r ( G ) as wcol r ( G ) := min π ∈ Π( G ) max v ∈ V ( G ) | WReach r [ G, π, v ] | . It is immediate from the definitions that col( G ) = wcol ( G ) ≤ wcol ( G ) ≤ . . . ≤ wcol n ( G ) = td( G ) . Hence, the weak r -coloring numbers can be seen as gradations between the coloring num-ber col( G ) and the treedepth td( G ) of G . The weak r -coloring numbers capture local separationproperties of G as follows. Lemma 19.
Let G be a graph, let π be an order of V ( G ) and let r ∈ N . Let u, v ∈ V ( G ) ,say u < π v , and assume that u WReach r [ G, π, v ] . Then every path P of length at most r connecting u and v intersects WReach r [ G, π, v ] ∩ WReach r [ G, π, u ] . Proof.
Let P be any path of length at most r connecting u and v . Then the minimum vertexof P lies both in WReach r [ G, π, v ] and in WReach r [ G, π, u ] . (cid:3) Theorem 20 ([38]).
A class C of graphs has bounded expansion if and only if for every r ∈ N there exists a number w = w ( r ) such that for every G ∈ C we have wcol r ( G ) ≤ w . Theorem 21 ([38, 27]).
A class C of graphs is nowhere dense if and only if for every r ∈ N and every real ǫ > there exists a number w = w ( r, ǫ ) such that for every H ⊆ G ∈ C we have wcol r ( H ) ≤ w · | V ( H ) | ǫ . To get used to the weak coloring numbers let us make the connection with the splitter game.
Theorem 22 ([21]).
Let G be a graph, let r ∈ N and let ℓ = wcol r ( G ) . Then splitter wins the ℓ -round radius- r splitter game on G . Proof.
Let π be a linear order with WReach r [ G, π, v ] ≤ ℓ for all v ∈ V ( G ) . Suppose in round i + 1 ≤ ℓ , connector chooses a subgraph C i +1 of G i of radius at most r . Let w i +1 (splitter’schoice) be the minimum vertex of C i +1 with respect to π . Then for each u ∈ V ( C i +1 ) there is apath between u and w i +1 of length at most r that uses only vertices of C i +1 . As w i is minimumin C i +1 , w i +1 is weakly r -reachable from each u ∈ V ( C i +1 ) . Now let G i +1 := C i +1 − { w i +1 } ] .As w i +1 is not part of G i +1 , in the next round splitter will choose another vertex which isweakly r -reachable from every vertex of the remaining graph. As WReach r [ G, π, v ] ≤ ℓ for all v ∈ V ( G ) , the game must stop after at most ℓ rounds. (cid:3) This gives for example a cubic number of rounds for splitter to win on planar graphs [37].Not surprisingly, the weak coloring numbers can also be used to give much improved bounds foruniform quasi-wideness on bounded expansion classes.
Theorem 23 ([24]).
Let G be a graph, A ⊆ V ( G ) , r, m ∈ N and assume wcol r ( G ) = c . If | A | ≥ · (2 cm ) c , then there exist sets S ⊆ V ( G ) and B ⊆ A \ S such that | S | ≤ c , | B | ≥ m ,and B is r -independent in G − S . We now come to the proof of Theorem 15, which follows from Theorem 21 and the followinglemma.
Lemma 24 ([19]).
Let G be a graph such that wcol r ( G ) ≤ s and let π be an order witnessingthis. For v ∈ V ( G ) , let m ( v ) be the minimum of N r ( v ) with respect to π . For each v ∈ V ( G ) let X r [ G, π, v ] := { w ∈ V ( G ) : v ∈ WReach r [ G, π, w ] } . Then X := { X r [ G, π, m ( v )] : v ∈ V ( G ) } is an r -neighborhood cover of G with radius at most r and maximum degree at most s . roof. Clearly the radius of each cluster is at most r , because if v is weakly r -reachablefrom w , then w ∈ N r ( v ) . Furthermore, for v ∈ V ( G ) we have N r ( v ) ⊆ X r [ G, π, m ( v )] . To seethis, let m ( v ) be the minimum of N r ( v ) with respect to π . Then m ( v ) is weakly r -reachable fromevery w ∈ N r ( v ) \{ m ( v ) } as there is a path from w to m ( v ) which uses only vertices of N r ( v ) andhas length at most r and m ( v ) is the minimum element of N r ( v ) . Thus N r ( v ) ⊆ X r [ G, π, m ( v )] .Finally observe that for every v ∈ V ( G ) , d X ( v ) = |{ u ∈ V ( G ) : v ∈ X r [ G, π, u ] }| = |{ u ∈ V ( G ) : u ∈ WReach r [ G, π, v ] }| = | WReach r [ G, π, v ] | ≤ s. (cid:3) Observe that the above defined neighborhood cover X of an n -vertex graph may have n elements, as there may be one cluster for every vertex. Hence, when branching over the elementsof the cover we may have a branching degree of n . However, the degree of the cover allows tobound the sum of all graphs in the recursion tree by O ( n ǫ ) for nowhere dense classes. Adifferent view on covers that leads to a smaller branching degree can be obtained as follows (wewould branch over the N subgraphs instead of over the n clusters). Theorem 25 ([32]).
Let G be a graph and let r ∈ N . Then there exist N ≤ w col r +1 ( G ) induced subgraphs H , . . . , H N of G such that1. for every v ∈ V ( G ) there is some ≤ i ≤ N with N r ( v ) ⊆ H i ;2. every connected component of the H i ’s has radius at most r . Proof.
Let π be a linear order of V ( G ) witnessing that wcol r +1 ( G ) ≤ N and let c : V ( G ) →{ , . . . , N } be a coloring so that c ( u ) = c ( v ) if u ∈ WReach r +1 [ G, π, v ] . Such a coloring canbe computed by a simple greedy procedure. Let H i be the subgraph of G induced by the sets X r [ G, π, u ] for all vertices u with c ( u ) = i , where X r [ G, π, u ] is defined as in Lemma 24. Letus show that the H i have the desired properties.As in the proof of Lemma 24 consider v ∈ V ( G ) and let m ( v ) be the minimum vertexof N r ( v ) with respect to π . Then m ( v ) is weakly r -reachable from every vertex in N r ( v ) thus N r ( v ) ⊆ H c ( m ( v )) and (1) holds.As observed before, for every v ∈ V ( G ) we have X r [ G, π, v ] ⊆ N r ( v ) . Now assume towardsa contradiction that there exist u < π u , z ∈ X r [ G, π, u ] , and z ∈ X r [ G, π, u ] such that c ( u ) = c ( u ) and z and z are either equal or adjacent. Then, considering a path of lengthat most r linking u and z with minimum u , the edge { z , z } if z = z and a path oflength at most r linking z and z with minimum u , we obtain a path of length at most r + 1 linking u and u with minimum u . Hence u is weakly (4 r +1) -reachable from u , contradictingthe hypothesis c ( u ) = c ( u ) . It follows that all connected components of H i are of the form X r [ G, π, v ] for some v ∈ V ( G ) hence have radius at most r . Thus (2) holds. (cid:3) Without going into more details: neighborhood covers can now be used to group verticesappropriately and to efficiently solve the model-checking problem. Further applications of theweak coloring numbers are in the efficient approximation of the distance- r dominating set prob-lem [2, 8, 9], as well as in the kernelization of distance- r dominating set and distance- r indepen-dent set [6, 11, 33]. Nowhere dense graph classes have a rich algorithmic theory and in particular, under the as-sumption of subgraph closure, these classes constitute the border of tractability for first-ordermodel-checking. Current research follows two lines to extend this border of tractability beyondsubgraph closed graph classes. The first line aims to study classes that are obtained as first-order10nterpretations or transductions of bounded expansion or nowhere dense classes. For exampleone obtains the class of map graphs as a first-order transduction from the class of planar graphs.Classes that are obtained as first-order transductions of sparse graph classes are called struc-turally sparse in [17]. It is a natural conjecture that good algorithmic properties of structurallysparse classes are inherited from the sparse base classes. I refer to [23, 16, 15, 17, 30] for progressin this direction.The second line of research is motivated by the observation that nowhere dense graph classesare monadically stable [1], a property that is studied in model theory, see e.g. [3]. Model theoryoffers a wealth of tools that could be exploited in an algorithmic context. For example, weproved in [12] that the distance- r dominating set problem is fixed-parameter tractable on everyclass of graphs where the distance- r formula is both stable and equational. References [1] H. Adler and I. Adler. Interpreting nowhere dense graph classes as a classical notion ofmodel theory.
Eur. J. Comb. , 36:322–330, 2014.[2] S. A. Amiri, P. Ossona de Mendez, R. Rabinovich, and S. Siebertz. Distributed domina-tion on graph classes of bounded expansion. In
Proceedings of the 30th on Symposium onParallelism in Algorithms and Architectures, SPAA 2018 , pages 143–151. ACM, 2018.[3] J. T. Baldwin and S. Shelah. Second-order quantifiers and the complexity of theories.
NotreDame Journal of Formal Logic , 26(3):229–303, 1985.[4] J. Chen, X. Huang, I. A. Kanj, and G. Xia. Strong computational lower bounds via param-eterized complexity.
Journal of Computer and System Sciences , 72(8):1346–1367, 2006.[5] A. Dawar, M. Grohe, and S. Kreutzer. Locally excluding a minor. In
Proceedings ofLICS 2007 , pages 270–279, 2007.[6] P. G. Drange, M. S. Dregi, F. V. Fomin, S. Kreutzer, D. Lokshtanov, M. Pilipczuk,M. Pilipczuk, F. Reidl, F. S. Villaamil, S. Saurabh, S. Siebertz, and S. Sikdar. Kernel-ization and sparseness: the case of dominating set. In , volume 47 of
LIPIcs , pages 31:1–31:14. SchlossDagstuhl - Leibniz-Zentrum fuer Informatik, 2016.[7] Z. Dvořák. Asymptotical structure of combinatorial objects. 2007.[8] Z. Dvorak. Constant-factor approximation of the domination number in sparse graphs.
Eur.J. Comb. , 34(5):833–840, 2013.[9] Z. Dvořák. On distance-dominating and-independent sets in sparse graphs.
Journal ofGraph Theory , 91(2):162–173, 2019.[10] Z. Dvořák, D. Král, and R. Thomas. Deciding first-order properties for sparse graphs. In
Proceedings of FOCS 2010 , pages 133–142, 2010.[11] K. Eickmeyer, A. C. Giannopoulou, S. Kreutzer, O. Kwon, M. Pilipczuk, R. Rabinovich,and S. Siebertz. Neighborhood complexity and kernelization for nowhere dense classesof graphs. In , volume 80 of
LIPIcs , pages 63:1–63:14. Schloss Dagstuhl - Leibniz-Zentrumfuer Informatik, 2017. 1112] G. Fabianski, M. Pilipczuk, S. Siebertz, and S. Toruńczyk. Progressive algorithms fordomination and independence. In , volume 126 of
LIPIcs , pages 27:1–27:16. Schloss Dagstuhl- Leibniz-Zentrum fuer Informatik, 2019.[13] J. Flum and M. Grohe. Fixed-parameter tractability, definability, and model-checking.
SIAM Journal on Computing , 31(1):113–145, 2001.[14] M. Frick and M. Grohe. Deciding first-order properties of locally tree-decomposable struc-tures.
Journal of the ACM , 48(6):1184–1206, 2001.[15] J. Gajarský, P. Hlinený, J. Obdrzálek, D. Lokshtanov, and M. S. Ramanujan. A newperspective on FO model checking of dense graph classes. In
Proceedings of the 31st AnnualACM/IEEE Symposium on Logic in Computer Science, LICS 2016 , pages 176–184. ACM,2016.[16] J. Gajarský and D. Král. Recovering sparse graphs. In , volume 117 of
LIPIcs , pages29:1–29:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018.[17] J. Gajarský, S. Kreutzer, J. Nesetril, P. Ossona de Mendez, M. Pilipczuk, S. Siebertz, andS. Toruńczyk. First-order interpretations of bounded expansion classes. In , volume 107of
LIPIcs , pages 126:1–126:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018.[18] M. Grohe, S. Kreutzer, R. Rabinovich, S. Siebertz, and K. Stavropoulos. Colouring andcovering nowhere dense graphs. In
International Workshop on Graph-Theoretic Conceptsin Computer Science , pages 325–338. Springer, 2015.[19] M. Grohe, S. Kreutzer, and S. Siebertz. Deciding first-order properties of nowhere densegraphs.
Journal of the ACM (JACM) , 64(3):17, 2017.[20] S. Kreutzer. Algorithmic meta-theorems. In
Finite and Algorithmic Model Theory , LondonMathematical Society Lecture Note Series, chapter 5, pages 177–270. Cambridge UniversityPress, 2011.[21] S. Kreutzer, M. Pilipczuk, R. Rabinovich, and S. Siebertz. The generalised colouring num-bers on classes of bounded expansion. In , volume 58 of
LIPIcs , pages 85:1–85:13.Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016.[22] S. Kreutzer, R. Rabinovich, and S. Siebertz. Polynomial kernels and wideness properties ofnowhere dense graph classes.
ACM Transactions on Algorithms (TALG) , 15(2):24, 2018.[23] O. Kwon, M. Pilipczuk, and S. Siebertz. On low rank-width colorings. In
Graph-TheoreticConcepts in Computer Science - 43rd International Workshop, WG 2017 , volume 10520 of
Lecture Notes in Computer Science , pages 372–385. Springer, 2017.[24] W. Nadara, M. Pilipczuk, R. Rabinovich, F. Reidl, and S. Siebertz. Empirical evaluationof approximation algorithms for generalized graph coloring and uniform quasi-wideness.In , volume 103 of
LIPIcs , pages 14:1–14:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018.[25] J. Nešetřil and P. Ossona de Mendez. Tree-depth, subgraph coloring and homomorphismbounds.
European Journal of Combinatorics , 27(6):1022–1041, 2006.1226] J. Nešetřil and P. Ossona de Mendez. Grad and classes with bounded expansion I. decom-positions.
European Journal of Combinatorics , 29(3):760–776, 2008.[27] J. Nešetřil and P. Ossona de Mendez. On nowhere dense graphs.
European Journal ofCombinatorics , 32(4):600–617, 2011.[28] J. Nešetril and P. Ossona de Mendez. Sparsity: Graphs, structures, and algorithms, volume28 of algorithms and combinatorics, 2012.[29] J. Nešetřil and P. Ossona de Mendez. Structural sparsity.
Russian Mathematical Surveys ,71(1):79, 2016.[30] J. Nesetril, P. Ossona de Mendez, R. Rabinovich, and S. Siebertz. Classes of graphswith low complexity: the case of classes with bounded linear rankwidth. arXiv preprintarXiv:1909.01564 , 2019.[31] J. Nešetřil, P. Ossona de Mendez, and D. R. Wood. Characterisations and examples ofgraph classes with bounded expansion.
European Journal of Combinatorics , 33(3):350–373,2012.[32] P. Ossona de Mendez. Unpublished observation, 2016.[33] M. Pilipczuk and S. Siebertz. Kernelization and approximation of distance-r independentsets on nowhere dense graphs.
CoRR , abs/1809.05675, 2018.[34] M. Pilipczuk, S. Siebertz, and S. Toruńczyk. On the number of types in sparse graphs.In
Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science ,pages 799–808. ACM, 2018.[35] N. Robertson and P. D. Seymour. Graph minors: XVII. taming a vortex.
Journal ofCombinatorial Theory, Series B , 77(1):162–210, 1999.[36] D. Seese. Linear time computable problems and first-order descriptions.
MathematicalStructures in Computer Science , 6(6):505–526, 1996.[37] J. Van den Heuvel, P. Ossona de Mendez, D. Quiroz, R. Rabinovich, and S. Siebertz. Onthe generalised colouring numbers of graphs that exclude a fixed minor.
European Journalof Combinatorics , 66:129–144, 2017.[38] X. Zhu. Colouring graphs with bounded generalized colouring number.