aa r X i v : . [ m a t h . C O ] J u l On weighted sublinear separators
Zdenˇek Dvoˇr´ak ∗ Abstract
Consider a graph G with an assignment of costs to vertices. Evenif G and all its subgraphs admit balanced separators of sublinear size , G may only admit a balanced separator of sublinear cost after deletinga small set Z of exceptional vertices. We improve the bound on | Z | from O (log | V ( G ) | ) to O (log log . . . log | V ( G ) | ), for any fixed number ofiterations of the logarithm. A balanced separator in a graph G is a set C ⊆ V ( G ) such that eachcomponent of G − C has at most | V ( G ) | vertices. A hereditary class ofgraphs G has strongly sublinear separators if there exist c, ε > G ∈ G has a balanced separator of size at most c | V ( G ) | − ε . Fa-mously, planar graphs [10] and more generally graphs without a fixed graphas a minor [1], as well as many geometrically defined graph classes [12, 13]have strongly sublinear separators, and this property has important algo-rithmic applications [11].Building upon a result of Plotkin et al. [14], Dvoˇr´ak and Norin [6] es-tablished a connection to shallow minors. For an integer ℓ ≥
0, a depth- ℓ minor of a graph G is a graph obtained from a subgraph of G by contractingpairwise vertex-disjoint subgraphs, each of radius at most ℓ . The density ofa graph H is | E ( H ) | / | V ( H ) | , and we define ∇ ℓ ( G ) as the maximum densityof a depth- ℓ minor of G . For a function f : N → N , we say the expansionof G is bounded by f if ∇ ℓ ( G ) ≤ f ( ℓ ) for every integer ℓ ≥
0. We say thata class of graphs G has polynomial expansion if there exists a polynomial p bounding the expansion of all graphs in G . Theorem 1 (Dvoˇr´ak and Norin [6]) . A hereditary class G of graphs hasstrongly sublinear separators if and only if it has polynomial expansion. ∗ Computer Science Institute, Charles University, Prague, Czech Republic. E-mail: [email protected] . Supported by the ERC-CZ project LL2005 (Algorithms andcomplexity within and beyond bounded expansion) of the Ministry of Education of CzechRepublic.
1n this note, we will focus on the weighted version of the separators.There are two senses in which one could interpret this statement: Theweights could affect the balance or the size of the separators. While weprimarily focus on the latter sense, we will need to use the former sensein the course of our argument as well. To avoid confusion, we say thatan assignment of non-negative real numbers to vertices is a weight assign-ment if the values are used to control the balance of the separators anda cost assignment if they are used to control the size of the separators.Given a weight or cost assignment f : V ( G ) → R +0 for a graph G and a set X ⊆ V ( G ), we define f ( X ) = P x ∈ X f ( x ), and for a subgraph H ⊆ G , wedefine f ( H ) = f ( V ( H )).Consider a cost assignment ρ : V ( G ) → R +0 for a graph G . Even if G belongs to a class with strongly sublinear separators, it may not necessarilyhave a balanced separator of small cost. For example, suppose G is a starwith center v and leaves v , . . . , v n , and let ρ ( v ) = n/ ρ ( v i ) = 1 for i = 1 , . . . , n . Any balanced separator C must contain either v or at least n/ ρ ( C ) ≥ n/ ≥ ρ ( G ) /
4. However, as in this example,it may be the case that a cheap separator exists after deleting a boundednumber of expensive vertices. More precisely, for integers t ≥ q ≥ C ⊆ V ( G ) is ( ρ/t ) -cheap with q outliers if there exists a set C ′ ⊆ C of size at most q such that ρ ( C \ C ′ ) ≤ ρ ( G ) /t . In [5], I conjecturedit is always possible to get a cheap balanced separator with only a boundednumber of outliers. Conjecture 1.
For every polynomial p , there exists a function q : N → N such that the following holds. Let G be a graph with expansion bounded by p and let ρ : V ( G ) → R +0 be a cost assignment for G . For every integer t ≥ , G has a balanced separator which is ( ρ/t ) -cheap with q ( t ) outliers. Let me remark that Conjecture 1 is a weakening of my conjecture thatgraphs with polynomial expansion are fractionally treewidth-fragile [4], andconsequently the conclusion of Conjecture 1 holds for graphs from every classknown to be fractionally treewidth-fragile. This includes all proper minor-closed classes [2] and all graphs with polynomial expansion and boundedmaximum degree [4]. Moreover, in [4], I proved the following weakening ofthis conjecture.
Theorem 2 (Dvoˇr´ak [5]) . For every polynomial p : N → N , there existsa polynomial q : N → N such that the following holds. Let G be an n -vertex graph with expansion bounded by p and let ρ : V ( G ) → R +0 be a costassignment for G . For every integer t ≥ , G has a balanced separator whichis ( ρ/t ) -cheap with q ( t log n ) outliers.
2s our main result, we make a further step towards Conjecture 1 by im-proving the dependence on the number of vertices. Let us define log (0) ( x ) = x and log ( i +1) x = log max (cid:0) , log ( i ) ( x ) (cid:1) for i ≥ Theorem 3.
For every polynomial p : N → N and integer a ≥ , thereexists a polynomial q : N → N such that the following holds. Let G be an n -vertex graph with expansion bounded by p and let ρ : V ( G ) → R +0 be acost assignment for G . For every integer t ≥ , G has a balanced separatorwhich is ( ρ/t ) -cheap with exp( q ( t )) log ( a ) n outliers. Note that unlike Theorem 2, the bound in Theorem 3 is exponential in t , and thus it beats Theorem 2 only for t = o (log log n ). Our approach givesa better bound in terms of weak coloring numbers. Let G be a graph andlet ≺ be a linear ordering of its vertices. For a vertex v ∈ V ( G ) and aninteger r ≥
0, a vertex u is ( ≺ , r ) -reachable from v if there exists a path P from v to u in G of length at most r such that u (cid:22) x for every x ∈ V ( P )(and in particular, u (cid:22) v ). Let L ≺ r ( v ) denote the set of vertices ( ≺ , r )-reachable from v , and let us define wcol ≺ r ( G ) = max {| L ≺ r ( v ) | : v ∈ V ( G ) } .The weak r -coloring number wcol r ( G ) is the minimum of wcol ≺ r ( G ) over alllinear orderings ≺ of V ( G ). Theorem 4.
For every polynomial p : N → N and integer a ≥ , thereexists a polynomial q : N → N such that the following holds. Let G be an n -vertex graph with expansion bounded by p and let ρ : V ( G ) → R +0 be acost assignment for G . For every integer t ≥ , G has a balanced separatorwhich is ( ρ/t ) -cheap with q ( t ) wcol q ( t ) ( G ) log ( a ) n outliers. Note that weak coloring numbers and the expansion of a graph are linked,as described in the following lemma which follows from known bounds (wegive a more detailed argument in the Appendix).
Lemma 5.
For every graph G and integer r ≥ , we have wcol r ( G ) ≤ (cid:16) r ∇ r ( G ) (cid:17) r and ∇ r ( G ) ≤ wcol r ( G ) . Hence, Theorem 4 implies Theorem 3. Let us remark that there existgraphs with polynomial expansion and superpolynomial weak coloring num-bers [8]. However, weak coloring numbers are known to be polynomial for anumber of interesting graph classes [15].Let us give two further interpretations of our main result. For an integer r ≥
1, we say that C ⊆ V ( G ) is a balanced distance- r separator in a graph G if V ( G − C ) = A ∪ B for disjoint sets A and B such that | A | , | B | ≤ | V ( G ) | and the distance d G ( A, B ) between A and B in G is greater than r . Note3hat stars do not have small balanced distance-2 separators. However, thiscan again be worked around by deleting a few vertices first. Theorem 6.
For every polynomial p : N → N and integers a, r ≥ , thereexists a polynomial q ′ : N → N such that the following holds. Let G be an n -vertex graph with expansion bounded by p . For every integer t ≥ , thereexists a set Z ⊆ V ( G ) of size at most exp( q ′ ( t )) log ( a ) n such that G − Z hasa balanced distance- r separator of size at most n/t . Theorem 6 is proved by applying Theorem 4 to a graph arising from theweak coloring number at distance r ; see the next section for more details.A more direct application is to balanced edge separators , sets of edgeswhose removal splits the graph into components with at most 2 / ρ ( v ) = deg v and including in the edge separator allthe edges incident with the non-outlier vertices, we obtain the followingresult. Corollary 7.
For every polynomial p : N → N and integer a ≥ , thereexists a polynomial q : N → N such that the following holds. Let G be an n -vertex graph with expansion bounded by p . For every integer t ≥ , thereexists a set Z ⊆ V ( G ) of size at most q ( t ) wcol q ( t ) ( G ) log ( a ) n such that G − Z has a balanced edge separator of size at most | E ( G ) | /t . We will prove Theorem 4 in the setting of separators whose balance isaffected by weights of vertices (this makes the result slightly more general,but additionally we need to use the weights in the proof). For a weightassignment w : V ( G ) → R +0 , a w -balanced separator in a graph G is a set C ⊆ V ( G ) such that each component K of G − C satisfies w ( K ) ≤ w ( G ).We define η ( G, t ) as the minimum integer q such that for every weight as-signment w and cost assignment ρ , there exists a w -balanced separator in G which is ( ρ/t )-cheap with q outliers.Let us remark that at least in this weighted setting, we cannot replace q ( t ) by a constant independent of t in the statement of Conjecture 1. For s ≥
3, let G s,n be the graph obtained from the complete bipartite graph K s,n withparts A and B by subdividing each edge s − ∇ r ( G s,n ) = O ( r ). Let w be the weight assignment for G s,n such that w ( y ) = 1 for all y ∈ B and w ( x ) = 0 for all other vertices x . Let ρ be the cost assignmentsuch that ρ ( x ) = n for x ∈ A , ρ ( x ) = s for x ∈ B , and ρ ( x ) = 1 otherwise.We have ρ ( G s,n ) = s n . Consider an optimal w -balanced separator C with4 < s outliers. For n ≫ s , all outliers belong to A and the non-outliers of C have cost at least ( s − q ) n/
3. If s n/t ≥ ( s − q ) n/
3, then q > = s − s /t .For s = t/
6, this implies q ≥ t/
12. Hence, there exist graphs G with linearexpansion and with η ( G, t ) = Ω( t ). Given a graph G , a linear ordering ≺ of its vertices, and an integer m ≥ G [ ≺ ,m ] denote the graph with vertex set V ( G ) and the edges uv for all v ∈ V ( G ) and u ∈ L ≺ m ( v ). We need a bound on the expansion of this graphin terms of the expansion of G . Lemma 8.
Let G be a graph, let ≺ be a linear ordering of its vertices, andlet m ≥ and r ≥ be integers. Then ∇ r (cid:0) G [ ≺ ,m ] (cid:1) ≤ (cid:0) wcol ≺ m ( G ) (cid:1) (2 m + 1) (2 r + 1) ∇ (2 m +1) r + m ( G ) + wcol ≺ m ( G ) . In particular, if the expansion of G is bounded by p ( r ) = c ( r + 1) k forsome c ≥ and k ≥ , then the expansion of G [ ≺ ,m ] is bounded by p ′ ( r ) = c ′ ( r + 1) k +2 for c ′ = 5 (cid:0) wcol ≺ m ( G ) (cid:1) (2 m + 1) k +2 c. Proof.
For every u ∈ V ( G ), let R ≺ m ( u ) be the subgraph of G induced by { v ∈ V ( G ) : u ∈ L ≺ m ( v ) } . Note that v ∈ V ( R ≺ m ( u )) if and only if G containsa path P from u to v of length at most r such that u (cid:22) x for every x ∈ V ( P ).All vertices of such a path also belong to V ( R ≺ m ( u )). Consequently, R ≺ m ( u ) isconnected and has radius at most m . Moreover, every edge uv ∈ E ( G [ ≺ ,m ] )satisfies R ≺ m ( u ) ∩ R ≺ m ( v ) = ∅ , and every vertex v ∈ V ( G ) belongs to V ( R ≺ m ( u ))for | L ≺ m ( v ) | ≤ wcol ≺ m ( G ) vertices u ∈ V ( G ). The bound on ∇ r (cid:0) G [ ≺ ,m ] (cid:1) follows by Lemma 3.10 of Har-Peled and Quanrud [9].For a set X ⊆ V ( G ), let G [ ≺ ,m ] /X denote the induced subgraph of G [ ≺ ,m ] with vertex set S x ∈ X L ≺ m ( x ); note that | V ( G [ ≺ ,m ] /X ) | ≤ wcol ≺ m ( G ) | X | . Lemma 9.
Let G be a graph, let ≺ be a linear ordering of its vertices,let m ≥ be an integer, and let X be a set of vertices of G . Let C be aset of vertices of G [ ≺ ,m ] /X . If x, y ∈ X belong to different components of G [ ≺ ,m ] /X − C , then d G − C ( x, y ) > m .Proof. Suppose for a contradiction there exists a path P of length at most m from x to y in G − C , and let z = min ≺ V ( P ). If say z = x , then P shows that xy ∈ G [ ≺ ,m ] , contradicting the assumption that x and y belong5o different components of G [ ≺ ,m ] /X − C . Hence, we have x = z = y . Thesubpaths of P from x and y to z show that z ∈ L ≺ m ( x ) ∩ L ≺ m ( y ), and thus xz, yz ∈ E ( G [ ≺ ,m ] /X ). Since x and y belong to different components of G [ ≺ ,m ] /X − C , it follows that z ∈ C , which is a contradiction.We are now ready to derive the distance version of our main result. Proof of Theorem 6.
By Lemma 5, there exists an integer c r such that everygraph G with expansion bounded by p satisfies wcol r ( G ) ≤ c r . By Lemma 8,there exists a polynomial p ′ such that for every such graph G , there existsa linear ordering ≺ of V ( G ) such that the expansion of G [ ≺ ,r ] is boundedby p ′ . Let q be the polynomial from Theorem 3 for the given a and with p ′ playing the role of p . Let us define q ′ ( t ) = q ( c r t ).Consider any graph G with expansion bounded by p and let ≺ be alinear ordering of V ( G ) such that the expansion of G [ ≺ ,r ] is bounded by p ′ . For u ∈ V ( G ), let R ( u ) = { v ∈ V ( G ) : u ∈ L ≺ m ( v ) } and let ρ ( u ) = | R ( u ) | . By Theorem 3 applied with t ′ = c r t playing the role of t , there existsets C, Z ⊆ V ( G ) such that | Z | ≤ exp( q ( t ′ )) log ( a ) n = exp( q ′ ( t )) log ( a ) n , ρ ( C ) ≤ ρ ( G ) /t ′ , and C ∪ Z is a balanced separator in G [ ≺ ,r ] . In particular,we can divide the components of G [ ≺ ,r ] − ( C ∪ Z ) into two parts A ′ and B of size at most n . Let C ′ = S u ∈ C R ( u ) and A = A ′ \ C ′ . We have | C ′ | ≤ ρ ( C ) ≤ ρ ( G ) /t ′ ≤ c r n/t ′ = n/t .It remains to argue that C ′ is a balanced distance- r separator in G − Z .Indeed, suppose for a contradiction that P is a path of length at most r in G − Z from a vertex v ∈ A to a vertex v ∈ B , and let u = min ≺ V ( P ).Note that v uv is a path from v to v in G [ ≺ ,r ] , and since u Z and v and v belong to different components of G [ ≺ ,r ] − ( C ∪ Z ), we have u ∈ C .However, then v ∈ R ( u ) ⊆ C ′ . This is a contradiction, since v ∈ A . For integers t, b ≥
1, let ℓ ( t, b ) be the minimum integer ℓ such that (1 +1 /t ) ℓ > b . Since (1 + 1 /t ) t ≥
2, we have ℓ ( t, b ) ≤ t ⌈ log ( b + 1) ⌉ . Considera graph G with a weight assignment w and a cost assignment ρ . For aset X ⊆ V ( G ), let N G ( X ) denote the set of vertices in V ( G ) \ X with aneighbor in X . We say that G is a ( w, ρ, t ) -expander if ρ ( G ) = 0 and forevery X ⊆ V ( G ) such that w ( X ) ≤ w ( G ) /
2, we have ρ ( N G ( X )) ≥ ρ ( X ) /t . Lemma 10.
Let t, b , b ≥ be integers. Let G be a graph and let w and ρ be a weight and a cost assignment for G such that G is a ( w, ρ, t ) -expander. or any X , X ⊆ V ( G ) , if ρ ( X ) ≥ ρ ( G ) /b and ρ ( X ) ≥ ρ ( G ) /b , then d G ( X , X ) ≤ ℓ ( t, b ) + ℓ ( t, b ) .Proof. For i ≥ j ∈ { , } , let X ij denote the set of vertices of G atdistance at most i from X j . Suppose for a contradiction that d G ( X , X ) >ℓ ( t, b )+ ℓ ( t, b ), and thus X ℓ ( t,b )1 ∩ X ℓ ( t,b )2 = ∅ and w (cid:0) X ℓ ( t,b )1 (cid:1) + w (cid:0) X ℓ ( t,b gr )2 ) ≤ w ( G ). By symmetry, we can assume w (cid:0) X ℓ ( t,b )1 (cid:1) ≤ w ( G ) /
2. Since G is a( w, ρ, t )-expander, for 1 ≤ i ≤ ℓ ( t, b ), we have ρ ( X i ) ≥ (1 + 1 /t ) ρ ( X i − j ),and thus ρ ( X ℓ ( t,b )1 ) ≥ (1 + 1 /t ) ℓ ( t,b ) ρ ( X ) ≥ (1 + 1 /t ) ℓ ( t,b ) ρ ( G ) /b > ρ ( G ),by the definition of ℓ ( t, b ) and the assumption that ρ ( G ) = 0. This is acontradiction.For a graph G and an integer l ≥
1, let ω l ( G ) denote the largest cliquethat appears in G as a depth- l minor. Clearly, ω l ( G ) ≤ ∇ l ( G ) + 1. A tightdepth- l clique minor in G is a clique minor K of depth l where the vertex setof every bag is covered by at most ω l ( G ) − l with thesame starting point. Note this implies | V ( K ) | ≤ ω l ( G )(( ω l ( G ) − l + 1) ≤ ω l ( G ) l . For a positive integer n ′ , let η ≤ n ′ ( G, t ) denote the maximum of η ( H, t ) over all subgraphs H of G with at most n ′ vertices.Let us now prove a key result relating the number of outliers in a graph G to the number of outliers in a subgraph of G with polylogarithmic numberof vertices. Theorem 4 will follow by iterating this result. The argument isbased on the proof of Plotkin et al. [14], the novel idea being the way webreak up the expensive vertices in part (c). Theorem 11.
Let G be a graph with n vertices, let ≺ be a linear ordering ofthe vertices of G and let t ≥ be an integer. Let l = 2 ℓ (5 t, tn ) , b = ω l ( G ) l , m = 2 ℓ (5 t, t ) , r = ⌈ log / (20 t/ ⌉ , and n ′ = 5 bt wcol ≺ m ( G ) . Then η ( G, t ) ≤ rη ≤ n ′ ( G [ ≺ ,m ] , rt ) . Proof.
Let w and ρ be a weight and a cost assignment for G . We constructa sequence of tuples T i = ( A i , B i , C i , D i , K i , r i ) (for i = 0 , , . . . ), where A i , B i , C i , D i and K i are pairwise disjoint subsets of V ( G ) and r i ≥ i > A i − ∪ K i − ⊆ A i ∪ K i , B i − ⊆ B i , C i − ⊆ C i , D i − ⊆ D i , and r i − ≤ r i . Let us define R i = G − ( A i ∪ B i ∪ C i ∪ D i ∪ K i ),and let H i be the graph with vertex set { v ∈ V ( R i ) : ρ ( v ) > bt ρ ( G ) } andwith uv ∈ E ( H i ) if and only if d R i ( u, v ) ≤ m .We maintain the following invariants for every i ≥ w ( A i ) ≤ w ( G ) and N G ( A i ) ⊆ B i ∪ C i ∪ D i ∪ K i ,7ii) ρ ( A i ) ≥ tρ ( B i ),(iii) ρ ( C i ) ≤ r i /r + δ i t ρ ( G ), where δ i = 0 if V ( H i ) = ∅ and δ i = 1 otherwise,(iv) | D i | ≤ r i η ≤ n ′ ( G [ ≺ ,m ] , rt ),(v) K i is the vertex set of a tight depth- l clique minor such that ρ ( v ) ≤ bt ρ ( G ) for every v ∈ V ( K i ), and(vi) r i ≤ r and every component M of H i satisfies ρ ( M ) ≤ (8 / r i ρ ( G ).We let A = B = D = K = ∅ , r = 0, and C = { v ∈ V ( G ) : ρ ( v ) < tn ρ ( G ) } ; clearly, ρ ( C ) ≤ t ρ ( G ), and thus (iii) holds. All the other invari-ants are trivially satisfied.For i ≥
0, assuming we already determined ( A i , B i , C i , D i , K i ), we pro-ceed as follows.(a) If w ( R i ) ≤ w ( G ), the construction stops. By (i), B i ∪ C i ∪ D i ∪ K i is a w -balanced separator in G . By (ii), we have ρ ( B i ) ≤ t ρ ( A i ) ≤ t ρ ( G ).Since K i is a vertex set of a tight depth- l clique minor, we have | K i | ≤ b and ρ ( K i ) ≤ t ρ ( G ) by (v). Together with (iii) and (vi), this implies ρ ( B i ∪ C i ∪ K i ) ≤ ρ ( G ) /t . By (iv) and (vi), it follows that the set B i ∪ C i ∪ D i ∪ K i is ( ρ/t )-cheap with rη ≤ n ′ ( G [ ≺ ,m ] , rt ) outliers, asrequired.From now on, assume w ( R i ) > w ( G ). In particular, V ( R i ) = ∅ , andby the choice of C , ρ ( R i ) = 0.(b) If R i is not an ( w, ρ, t )-expander, then let Z i ⊆ V ( R i ) be such that w ( Z i ) ≤ w ( R i ) / ρ ( N R i ( Z i )) < t ρ ( Z i ). Let A i +1 = A i ∪ Z i , B i +1 = B i ∪ N R i ( Z i ), C i +1 = C i , D i +1 = D i , K i +1 = K i and r i +1 = r i .Note (ii) is satisfied by T i +1 , since it is satisfied by T i and ρ ( N R i ( Z i )) < t ρ ( Z i ). Moreover, w ( A i +1 ) ≤ w ( A i ) + w ( R i ) / ≤ ( w ( G ) − w ( R i )) + w ( R i ) / w ( G ) − w ( R i ) / < w ( G ) since w ( R i ) > w ( G ), and thus T i +1 satisfies (i). All the other invariants are clearly preserved.From now on, assume R i is a ( w, ρ, t )-expander.(c) Let us now consider the case V ( H i ) = ∅ . If ρ ( H i ) ≤ t ρ ( G ), then let T i +1 be obtained from T i by setting C i +1 = C i ∪ V ( H i ). By (iii), wehave ρ ( C i ) ≤ r i /r t ρ ( G ), and thus ρ ( C i +1 ) ≤ r i +1 /r t ρ ( G ). Moreover, V ( H i +1 ) = ∅ , and thus T i +1 satisfies (iii). All the other invariants areclearly preserved. Hence, suppose that ρ ( H i ) > t ρ ( G ).8et M i be a component of H i with ρ ( M i ) maximum. We claim that ρ ( M i ) ≥ ρ ( H i ). Indeed, suppose for a contradiction ρ ( M i ) < ρ ( H i ).Then we can express H i as a disjoint union of graphs H ′ i and H ′′ i such that ρ ( H ′ i ) , ρ ( H ′′ i ) > ρ ( H i ) / > t ρ ( G ). Since R i is a ( w, ρ, t )-expander, Lemma 10 implies d R i ( V ( H ′ i ) , V ( H ′′ i )) ≤ m . But then twovertices of R i at distance at most m from each other belong to differentcomponents of H i , contradicting the definition of H i .Therefore, we have ρ ( M i ) > t ρ ( G ), and by (vi), (8 / r i > t , im-plying r i < log / (20 t/ ≤ r .Let G ′ i = R [ ≺ ,m ] /V ( H i ) i , and let w i be the weight assignment defined by w i ( v ) = ρ ( v ) for v ∈ V ( H i ) and w i ( v ) = 0 otherwise. Note that G ′ i ⊆ G [ ≺ ,m ] . Moreover, we have ρ ( v ) > bt ρ ( G ) for each v ∈ V ( H i ), andthus | V ( H i ) | ≤ bt and V ( G ′ i ) ≤ | V ( H i ) | wcol ≺ m ( G ) ≤ bt wcol ≺ m ( G ) = n ′ . Therefore, G ′ i contains a w i -balanced separator X i ∪ Y i , where ρ ( X i ) ≤ rt ρ ( G ′ i ) and | Y i | ≤ η ≤ n ′ ( G [ ≺ ,m ] , rt ). Let T i +1 be obtainedfrom T i by setting C i +1 = C i ∪ X i , D i +1 = D i ∪ Y i , and r i +1 = r i + 1.Clearly, all the invariants except possibly for (vi) are satisfied.Let us now argue that (vi) holds. Since r i < r , we have r i +1 ≤ r .Note that R i +1 = R i − ( X i ∪ Y i ). By Lemma 9, if vertices u, v ∈ V ( H i ) \ ( X i ∪ Y i ) belong to different components of G ′ i − ( X i ∪ Y i ),then d R i +1 ( u, v ) > m , and thus uv E ( H i +1 ). Consequently, anycomponent M of H i +1 is contained in a component of G ′ i − ( X i ∪ Y i ).Since X i ∪ Y i is a w i -balanced separator in G ′ i and ρ ( M i ) ≥ ρ ( H i ), wehave ρ ( M ) = w i ( M ) ≤ w i ( G ′ i ) = ρ ( H i ) ≤ ρ ( M i ) ≤ (8 / r i +1 ρ ( G )by (vi). This implies that T i +1 satisfies (vi).From now on, we assume V ( H i ) = ∅ .(d) If there exists a bag S of the clique minor K i with no neighbor in R i , let T i +1 be obtained from T i by setting A i +1 = A i ∪ S and K i +1 = K i \ S .We have w ( A i +1 ) ≤ w ( G ) − w ( R i ) < w ( G ) /
3, and thus T i +1 satisfies(i). All other invariants are clearly satisfied as well.(e) Let S , . . . , S o be the bags of K i . By the previous paragraph, we canfor j = 1 , . . . , o assume that S j has a neighbor v j ∈ V ( R i ). Fix anyvertex v ∈ V ( R i ). By the choice of C , we have ρ ( v ) , ρ ( v j ) ≥ tn ρ ( G ).Since R i is a ( w, ρ, t )-expander, Lemma 10 implies that d R i ( v, v j ) ≤ l .Let S be the union of paths of length at most l from v to the vertices v ,. . . , v o in R i . Let T i +1 be obtained from T i by setting K i +1 = K i ∪ S .9learly, (v) holds, since V ( H i ) = ∅ , and thus ρ ( u ) ≤ bt ρ ( G ) for every u ∈ V ( R i ).Note that after each of the operations (b), (c), (d), and (e), the pair ( | A i +1 | + | C i +1 | + r i +1 , | K i +1 | ) is lexicographically strictly greater than ( | A i | + | C i | + r i , | K i | ), and since | A j | + | C j | + r j ≤ n + r and | K j | ≤ n for each j , theprocess necessarily stops after a finite number of steps.We now iterate Theorem 11 a times and use the trivial bound η ( H, t ) ≤| V ( H ) | at the end. Let us remark that for r , r ≥
1, a graph G , and alinear ordering ≺ of the vertices of G , if H is a subgraph of G [ ≺ ,r ] , then H [ ≺ ,r ] ⊆ G [ ≺ ,r r ] . Corollary 12.
Let G be a graph with n vertices, let ≺ be a linear orderingof the vertices of G and let t ≥ be an integer. Let t = t , n = n , m = 1 and for i ≥ , let • r i = ⌈ log / (20 t i / ⌉ , t i +1 = 5 r i t i , • m i +1 = 2 ℓ (5 t i , t i ) m i , • l i = 2 ℓ (5 t i , t i n i ) , b i = ω l i ( G [ ≺ ,m i ] ) l i , n i +1 = 5 b i t i wcol ≺ m i +1 ( G ) .For every a ≥ , we have η ( G, t ) ≤ n a Q a − i =0 r i . Let us estimate the quantities from Corollary 12. Let p ( r ) = c ( r + 1) k be a polynomial bounding the expansion of G . We consider p as well as thenumber of iterations a to be fixed, and thus we hide multiplicative termsdepending only on them in the O -notation. By Lemma 8, for every l, m ≥ ω l ( G ≺ ,m ) = O ( ∇ l ( G ≺ ,m )) = O (cid:16)(cid:0) wcol ≺ m ( G ) (cid:1) ( ml ) O (1) (cid:17) . We have r i = O (log t i ), and thus t i = O ( t log i t ) for i ≤ a . Hence, m i +1 = O ( m i t i log t i ) = O ( m i t log i +1 t ), and thus m i = O ( t O (1) ) for i ≤ a .Note that n i +1 is a function of l i = O ( t i log t i n i ), and the logarithmdiminishes the effects of the earlier iterations. Hence, we can with no greatloss use the value of m i and t i from the last iteration in all previous iterationsas well. Moreover, we can choose the ordering ≺ so that wcol ≺ m a ( G ) =wcol m a ( G ). Let s = wcol m a ( G ), so that ω l i (cid:0) G ≺ ,m a (cid:1) = O (cid:0) s ( m a l i ) O (1) (cid:1) . We10ave n i +1 = O (cid:16) ω l i (cid:0) G [ ≺ ,m a ] (cid:1) l i t a s (cid:17) = O (cid:0) s m O (1) a t a l O (1) i (cid:1) = O (cid:16) s (cid:0) t log max(2 , n i ) (cid:1) O (1) (cid:17) . (1)By Lemma 5, we have log s = O ( m a log p ( m a )) = O ( m a log m a ) = O ( t O (1) ).Hence, an inductive argument using (1) implies n a = O (cid:16) s (cid:0) t log ( a ) n (cid:1) O (1) (cid:17) . Corollary 13.
For every polynomial p : N → N and integer a ≥ , thereexists a polynomial q : N → N such that the following holds. For every graph G and an integer t ≥ , if G has expansion bounded by p , then η ( G, t ) ≤ wcol q ( t ) ( G ) q (cid:0) t log ( a ) n (cid:1) . Theorem 4 follows from Corollary 13; we use the fact that q (log ( a +1) n ) = O (log ( a ) n ) to make the dependency on log ( a ) n linear. References [1]
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Appendix
Let G be a graph and let ≺ be a linear ordering of its vertices. For a vertex v ∈ V ( G ) and an integer r ≥
0, let κ ≺ r ( v ) denote the maximum number of12aths of length at most r in G starting in v , pairwise disjoint except for v ,and ending in { x ∈ V ( G ) : x ≺ v } . Let adm ≺ r ( G ) = max { κ ≺ r ( v ) : v ∈ V ( G ) } .The r -admissibility adm r ( G ) of G is the minimum of adm ≺ r ( G ) over all linearorderings ≺ of V ( G ). By a detour via another notion (strong r -coloringnumber), it is easy to see that wcol ≺ r ( G ) ≤ (cid:16) adm ≺ r ( G ) (cid:17) r , see e.g. [3] fordetails. However, a better bound follows by a direct argument. Lemma 14.
Let G be a graph and let ≺ be a linear ordering of its vertices.Then for every r ≥ , we have wcol ≺ r ( G ) ≤ (cid:16) r adm ≺ r ( G ) (cid:17) r . Proof.
Let ~H be the auxiliary directed graph with vertex set V ( G ) ×{ , . . . , r } ,edges (( u, i ) , ( v, i + 1)) and (( v, i ) , ( u, i + 1)) for each uv ∈ E ( G ) and 0 ≤ i ≤ r −
1, and edges ( v, i ) , ( v, i + 1) for each v ∈ V ( G ) and 0 ≤ i ≤ r − v, i ) ∈ V ( ~H ), let π ( v, i ) = v , and for a path Q in ~H , let m ( Q ) = min ≺ π ( V ( Q )).Consider any vertex v ∈ V ( G ) and let ~T be a minimal subgraph of ~H containing for each u ∈ L ≺ r ( v ) a path P from ( v,
0) to ( u, r ) such that m ( P ) = u . We claim that ~T has maximum indegree at most one. Indeed, forany vertex x ∈ V ( ~T ), there must by the minimality of ~T exist a path Q in ~T from ( v,
0) to x . Choose such a path Q with m ( Q ) maximum, and let e bethe last edge of Q . If an edge e ′ = e entered x , then ~T − e ′ would contradictthe minimality of ~T , since in any path P from ( v,
0) in ~T containing theedge e ′ , we can replace the initial segment by Q without decreasing m ( P ).Therefore, ~T is an outbranching. Furthermore, consider any vertex x ∈ ~T , and let P , . . . , P t be paths in ~T from x to the leaves of ~T starting withpairwise different edges. By the minimality of ~T , for 1 ≤ i ≤ t , the path P i has length at most r and satisfies m ( P i ) ≺ π ( x ). Moreover, the paths P − x ,. . . , P t − x are pairwise vertex-disjoint. Let F be an auxiliary graph withvertex set { , . . . , t } and with ij ∈ E ( F ) if and only if π ( P i − x ) ∩ π ( P j − x ) = ∅ . Note that F has maximum degree at most r ( r − ≤ r −
1, and thus F contains an independent set of size at least t/r . On the other hand, α ( F ) ≤ κ ≺ r ( π ( x )) ≤ adm ≺ r ( G ). Hence, the maximum outdegree of ~T is atmost r adm ≺ r ( G ). Consequently, the number of leaves of ~T , which equals | L ≺ r ( v ) | , is at most (cid:16) r adm ≺ r ( G ) (cid:17) r .Grohe et al. [8] proved the following bound on the admissibility. Theorem 15 (Grohe et al. [8], a consequence of Theorem 3.1) . For everygraph G and integer r ≥ , we have adm r ( G ) ≤ r ∇ r ( G ) ..