VVC-dimension and Erd˝os-P´osa property
Nicolas Bousquet , St´ephan Thomass´e a Universit´e Montpellier 2, 161 rue Ada, 34392 Montpellier Cedex 5 - France. b LIP, ENS Lyon, 46 all´ee d’Italie, 69364 Lyon Cedex 07 - France.
Abstract
Let G = ( V, E ) be a graph. A k -neighborhood in G is a set of vertices consisting of all the vertices atdistance at most k from some vertex of G . The hypergraph on vertex set V which edge set consists ofall the k -neighborhoods of G for all k is the neighborhood hypergraph of G . Our goal in this paper is toinvestigate the complexity of a graph in terms of its neighborhoods. Precisely, we define the distance VC-dimension of a graph G as the maximum taken over all induced subgraphs G (cid:48) of G of the VC-dimensionof the neighborhood hypergraph of G (cid:48) . For a class of graphs, having bounded distance VC-dimension bothgeneralizes minor closed classes and graphs with bounded clique-width.Our motivation is a result of Chepoi, Estellon and Vax`es [5] asserting that every planar graph of diameter2 (cid:96) can be covered by a bounded number of balls of radius (cid:96) . In fact, they obtained the existence of a function f such that every set F of balls of radius (cid:96) in a planar graph admits a hitting set of size f ( ν ) where ν is themaximum number of pairwise disjoint elements of F .Our goal is to generalize the proof of [5] with the unique assumption of bounded distance VC-dimensionof neighborhoods. In other words, the set of balls of fixed radius in a graph with bounded distance VC-dimension has the Erd˝os-P´osa property. Keywords: dominating set, distance VC-dimension, Erd˝os-P´osa property, clique-minor, rankwidth.
1. Introduction B -hypergraph and dominating sets. Let G = ( V, E ) be a graph. A dominating set of G is a set X of verticessuch that for every vertex v , there exists a vertex x ∈ X satisfies either x = v or v is a neighbor of x . Inother words, all the vertices of V are at distance at most one from a vertex of X . In this paper we focuson a generalization of dominating sets called dominating sets at distance (cid:96) . A set X is a dominating set atdistance (cid:96) if every vertex of the graph is at distance at most (cid:96) from a vertex of X .A hypergraph is a pair ( V, F ) where V is a set of vertices and F is a set of subsets of V called hyperedges.For the study of dominating sets, a natural hypergraph arises: the B -hypergraph. The B -hypergraph of G has vertex set V and hyperedges are the closed neighborhoods of the vertices of the graph. Since weconsider neighborhoods at distance (cid:96) in this paper, we naturally generalize the B -hypergraph into the B (cid:96) -hypergraph by replacing closed neighborhoods by balls of radius (cid:96) centered in every vertex of the graph. The B -hypergraph is the edge-union of the B (cid:96) -hypergraphs for all (cid:96) .A hitting set of a hypergraph H = ( V, F ) is a subset of vertices intersecting every hyperedge. In otherwords, it is a subset X of vertices such that for every e ∈ F , e ∩ X (cid:54) = ∅ . One can note that a hitting setof the B (cid:96) -hypergraph of a graph G is a dominating set at distance (cid:96) of the graph G (and the converse alsoholds). Indeed, let X be a hitting set of the B (cid:96) -hypergraph H of G . For every vertex v ∈ V , there exists x ∈ X such that x is at distance at most (cid:96) from v . So the whole set of vertices is at distance at most (cid:96) froma vertex of X , i.e. X is a dominating set at distance (cid:96) of G . In the following we focus on hitting sets of the [email protected] fi[email protected]. Partially supported by ANR Project STINT under Contract ANR-13-BS02-0007. Preprint submitted to Elsevier October 9, 2018 a r X i v : . [ m a t h . C O ] D ec igure 1: A shattered set of size 3. B (cid:96) -hypergraphs. The minimum size of a hitting set, denoted by τ , is called the transversality . The packingnumber , denoted by ν , is the maximum number of pairwise disjoint hyperedges. Complexity of graphs and VC-dimension.
A set X of vertices is shattered (resp. 2 -shattered ) if for everysubset X (cid:48) of X (resp. every subset of X of size 2) there exists a hyperedge e such that e ∩ X = X (cid:48) (seeFigure 1). Introduced in [18, 19], the Vapnik-Chervonenkis dimension (or
VC-dimension for short) (resp.2
VC-dimension ) of a hypergraph H is the maximum size of a shattered set (resp. 2-shattered set). It is agood complexity measure of a hypergraph, for instance in the learnability sense. A bounded VC-dimensionprovides upper bounds on the number of hyperedges [18] but also on the transversality [6, 13, 16]. TheVC-dimension has many applications, in learnability theory [13] and in computational geometry [4]. Morerecently, several applications were developed in graph theory, see [1, 5, 14] for instance.One of our goals was to extend this notion on graphs to catch the complexity of a graph at large distance.The distance VC-dimension of a graph G could be defined as the VC-dimension of the B -hypergraph of thegraph G . Since throughout this paper we only consider graphs closed under induced subgraphs, we definethe distance VC-dimension of a graph G (resp. distance VC-dimension of the graph G ) is the maximumover induced subgraphs of the distance VC-dimension (resp. 2VC-dimension) of the B -hypergraph. Sincethe VC-dimension “measures” the local randomness of hypergraphs, it is natural to think that classes witha lot of structure might have a bounded VC-dimension. In Section 3, we prove that two famous graphclasses have bounded distance VC-dimension. First we show that the class of K n -minor free graphs hasdistance VC-dimension at most n −
1. The proof is almost the proof of Chepoi, Estellon and Vax`es thatthe B (cid:96) -hypergraph of planar graphs has distance VC-dimension at most 4 . Then we show that the classof bounded rankwidth graphs have bounded distance VC-dimension. Actually, we prove a slightly strongerstatement for these two classes: their distance 2VC-dimension is bounded. We finally provide some graphsof bounded distance VC-dimension with an arbitrarily large distance 2VC-dimension. Erd˝os-P´osa property.
Chepoi, Estellon and Vax`es [5] proved that every planar graph of diameter 2 (cid:96) can becovered by c balls of radius (cid:96) (where c does not depend on (cid:96) ). It answered a conjecture of Gavoille, Peleg,Raspaud and Sopena [11]. Their proof uses the concept of VC-dimension but also planarity of the graph.One of our aims was to determine if the planarity arguments are necessary or if a purely combinatorial proofof this result exists.Let G be a graph. We denote by respectively ν (cid:96) and τ (cid:96) the packing number and the transversality of the B (cid:96) -hypergraph of G . Note that the B (cid:96) -hypergraph of a planar graph of diameter 2 (cid:96) satisfies ν (cid:96) = 1. Indeedfor every u, v ∈ V , since the diameter of the graph is at most 2 (cid:96) , there exists a vertex x at distance at most (cid:96) from both u and v , so the hyperedges centered in u and in v intersect. Since τ (cid:96) equals the minimum size ofa dominating set at distance (cid:96) , we have τ (cid:96) ≥ ν (cid:96) . A class of hypergraphs such that the transversality of everyhypergraph is bounded by a function of its packing number is said to satisfy the Erd˝os-P´osa property (andthe function is called the gap function ). In their seminal paper [9], Erd˝os and P´osa proved that the minimumsize of a feedback vertex set can be bounded by a function of the the maximum number of vertex disjoint In their paper, Chepoi, Estellon and Vax`es noted that their proof for planar graphs can be extended to K n -minor freegraphs. G has the erd˝os-P´osa property (the vertices of the hypergraphare the vertices of the graph and the hyperedges are the cycles of the graph).In Section 4, we first simplify and generalize the proof of Chepoi, Estellon and Vax`es. More preciselywe prove that the B (cid:96) -hypergraph of any graph G has a dominating set at distance (cid:96) of size O ( ν d +1 (cid:96) ) where d denotes the distance 2VC-dimension of G . Note that the function depends on ν (cid:96) but not directly on (cid:96) .Since planar graphs have distance 2VC-dimension at most 4, it ensures that the B (cid:96) hypergraph of any planargraph of diameter 2 (cid:96) satisfies τ (cid:96) ≤ f such that the B (cid:96) -hypergraph of a graphof distance VC-dimension d has a hitting set of size at most f ( ν (cid:96) , d ). The original proof of Chepoi, Estellonand Vax`es for planar graphs is based on the same method but they conclude using topological propertiesof planar graphs. Since we only deal with combinatorial structures, our proof is more technically involved.Note nevertheless that the function f is exponential in the distance VC-dimension while the one providedby the distance 2VC-dimension is polynomial.We will finally close this paper by some concluding remarks and open problems on distance VC-dimensionand Erd˝os-P´osa property.
2. Preliminaries
It is sometimes convenient to see a hypergraph as its incidence bipartite graph B H with vertex set V ∪ E in which there is an edge between x ∈ V and e ∈ E iff x ∈ e . Note that the pair ( V, E ) is oriented,and the hypergraph associated to the pair (
E, V ) is called the dual hypergraph . The vertices of the dualhypergraph are the hyperedges of the original one, and the hyperedges of the dual hypergraph are thesubsets of E containing the vertex v , for every v . The dual VC-dimension of H is the VC-dimension ofthe dual hypergraph of H . The VC-dimension of H and the dual VC-dimension of H are equivalent upto an exponential function [3]. Similarly, the dual VC-dimension of H is the 2VC-dimension of the dualhypergraph of H . The 2VC-dimension is larger than or equal to the VC-dimension and the gap can bearbitrarily large. Indeed, consider the clique K n . Its 2VC-dimension is equal to n whereas its VC-dimensionis at most 2 (since no hyperedge contains 3 vertices). The same example ensures that no function links2VC-dimension and dual 2VC-dimension.A transversal set (or hitting set ) of a hypergraph H is a set of vertices intersecting each hyperedge.The transversality τ of a hypergraph is the minimum size of a transversal set. The packing number ν ofa hypergraph is the maximum number of vertex disjoint hyperedges. A class of hypergraphs H has the Erd˝os-P´osa property if there exists a function f such that for all H ∈ H , τ ≤ f ( ν ). We denote by ν (cid:96) and τ (cid:96) respectively the packing number and the transversality of the B (cid:96) -hypergraph of G . Note that the B (cid:96) -hypergraph of a graph G and its dual are the same since for every pair of vertices x, y , x ∈ B ( y, (cid:96) ) if andonly if y ∈ B ( x, (cid:96) ). So: Observation 1.
The B (cid:96) -hypergraph is isomorphic to the dual of the B (cid:96) -hypergraph. Let G = ( V, E ) be a graph. Let X ⊆ V . The graph induced by X is the graph on vertex set X whoseedges are edges of G with both endpoints in X . A walk of length k from x ∈ V to y ∈ V is a sequenceof vertices x = x , x , . . . , x k − , x k = y where x i x i +1 ∈ E for each 0 ≤ i ≤ k −
1. A path is a walk withpairwise distinct vertices. The vertices x and y are the endpoints of the walk. The x i x j -subpath is the path x i , x i +1 , . . . , x j . The neighbors of the vertex x i on the path are the vertices x i − and x i +1 whenever theyexist. A minimum path from x to y , also called minimum xy -path , is a path of minimum length from x to y . The distance between x and y , denoted by d ( x, y ) is the length of a minimum xy -path when such a pathexists and + ∞ otherwise. The distance between a set X and a set Y is the minimum for all x, y ∈ X × Y of3he distance between x and y . The ball of center x and radius k , denoted by B ( x, k ), is the set of vertices atdistance at most k from x . The neighbors of x , denoted by N ( x ) are the vertices of B ( x,
1) distinct from x .Let us conclude this section by an observation which ensures that we can restrict our study to connectedsubgraphs: Observation 2.
The distance VC-dimension of a non connected graph is the distance VC-dimension of themaximum of its connected components.
3. Graphs of bounded distance VC-dimension
In this section we prove that K n minor-free graphs and bounded rank-width graphs have bounded distance2VC-dimension. In addition we provide a class of graphs with arbitrarily large distance 2VC-dimension anddistance VC-dimension at most 18. K d -minor-free graphs have bounded distance VC-dimension A graph H is a minor of G if H can be obtained from G by contracting edges, deleting edges, anddeleting vertices. Theorem 4 is roughly Proposition 1 of [5]. Since our definitions and statements are slightlydifferent, we prove it for the sake of completeness. We first prove an easy lemma before stating the maintheorem of this section. Lemma 3. If z is on a minimum xy -path, the ball B ( z, d ( x, z )) is included in B ( y, d ( x, y )) .Proof. Since z is on a minimum xy -path, d ( x, y ) = d ( x, z ) + d ( z, y ). Hence B ( y, d ( y, z )) contains z and then B ( y, d ( y, z ) + d ( z, x )) contains B ( z, d ( x, z )). Theorem 4. A K d -minor-free graph has distance VC-dimension at most d − .Proof. Let G be a graph with distance 2VC-dimension d . Let X = { x , x , . . . , x d } be a set of vertices of G which is 2-shattered by the hyperedges of the B -hypergraph of G . Hence, for every pair ( i, j ), there existsa vertex c i,j and an integer r i,j such that B ( c i,j , r i,j ) ∩ X = { x i , x j } . We assume moreover that r i,j isminimum for all choices of ( c i,j , r i,j ). A central path P i,j is the concatenation of a minimum path from x i to c i,j and a minimum path from c i,j to x j . Claim 1.
A central path is indeed a path.Proof.
Assume by contradiction that x appears more than once in a central path P i,j . Since P i,j is aconcatenation of a shortest x i c i,j -path and a shortest c i,j x j -path, x appears once between x i and c i,j andonce between c i,j and x j . Let us call Q the subpath of P i,j from x to c i,j and Q the subpath of P i,j from c i,j to x . Note that Q and Q are both shortest paths connecting c i,j and x , hence replacing Q bythe mirror of Q gives another central path P (cid:48) i,j . The two neighbors of c i,j in P (cid:48) i,j are the same vertex v ,contradicting the minimality of r i,j since B ( v, r i,j − ∩ X = { x i , x j } . Claim 2. If x belongs to two distinct central paths, then these paths are P i,j and P i,l , and we both have d ( x, x i ) < d ( x, x j ) and d ( x, x i ) < d ( x, x l ) .Proof. Assume that x appears in P i,j and P k,l , where d ( x, x i ) ≤ d ( x, x j ) and d ( x, x k ) ≤ d ( x, x l ). Freeto exchange the roles of P i,j and P k,l , we can also assume that d ( x, x k ) ≤ d ( x, x i ). By Lemma 3, x k ∈ B ( c i,j , r i,j ), hence we have x k = x i or x k = x j . Since d ( x, x k ) ≤ d ( x, x i ) ≤ d ( x, x j ) and x k is either x i or x j , we have d ( x, x k ) = d ( x, x i ). Hence d ( x, x i ) ≤ d ( x, x k ), and by the same argument, we have x i = x k or x i = x l . Since the central paths are distinct, we necessarily have x i = x k . Observe that d ( x, x i ) = d ( x, x j ),hence d ( x, x j ) ≤ d ( x, x i ), would give by the same argument x j = x k , hence a contradiction since we wouldhave x i = x j . Therefore d ( x, x i ) < d ( x, x j ), and for the same reason d ( x, x i ) < d ( x, x l ).4et us now construct some connected subsets X i for all 1 ≤ i ≤ d . For every path P i,j , the vertices of P i,j closer to x i than to x j are added to X i , the vertices of P i,j closer to x j than to x i are added to X j , andthe midvertex (if any) is arbitrarily added to X i or to X j .The crucial fact is that the sets X i are pairwise disjoint. Indeed, by Claim 1 and Claim 2, if a vertex x appears in two distinct central paths, these are P i,j and P i,l , where d ( x, x i ) < d ( x, x j ) and d ( x, x i ) < d ( x, x l ).In particular x belongs in both cases to X i .By construction, the sets X i are connected and there is always an edge between X i and X j since theirunion contains P i,j . Therefore if the distance 2VC-dimension is at least d , the graph contains K d as aminor. Let us first recall the definition of rankwidth, introduced by Oum and Seymour in [17]. Let G = ( V, E )be a graph and ( V , V ) be a partition of V . Let M V ,V be the matrix of size | V | × | V | such that the entry( x , x ) ∈ V × V equals 1 if x x ∈ E and 0 otherwise. The cutrank cr ( V , V ) of ( V , V ) is the rank of thematrix M V ,V over the field F . A ternary tree is a tree with nodes of degree 3 or 1. The nodes of degree 3are the internal nodes , the other nodes being the leaves . A tree-representation of G is a pair ( T, f ) where T is a ternary tree with | V | leaves and f is a bijection from V to the set of leaves. Every edge e of T defines apartition of the leaves of T . Therefore it defines a partition of the vertex set V into ( V e , V e ). The rankwidth rw of a graph G is defined by: rw ( G ) = min ( T,f ) max e ∈ E ( T ) cr ( V e , V e )Before stating the main result, let us first state two lemmas concerning rankwidth and ternary trees. Lemma 5.
Let G = ( V, E ) be a graph of rankwidth k and X, Y be the partition of V induced by an edge ofa tree-representation of G of cutrank k . There exist partitions of X and Y into at most k sets X , . . . , X k and Y , . . . , Y k such that for all i, j , ( X i × Y j ) ∩ E = ∅ or ( X i × Y j ) ∩ E = X i × Y j .Proof. Let T be a tree representation of G of cutrank at most k . Let e be an edge of the tree representationof G and ( X, Y ) be the partition of V induced by e . Since the cutrank is at most k , the matrix M X,Y hasrank at most k . Hence there exists j ≤ k rows R , . . . , R j which form a base of the rows of the matrix M X,Y .By definition, every row corresponds to the neighborhood of a vertex of X into Y . Let us denote by x i thevertex corresponding to R i . We denote by B the set { x , . . . , x j } .For every B (cid:48) ⊆ B , X ( B (cid:48) ) denotes the subset of X which contains x if N ( x ) ∩ Y = F (cid:80) x i ∈B (cid:48) N ( x i ). It inducesa partition of X since N ( x ) , . . . , N ( x j ) is a base of the neighborhoods of X in Y . Note that by definitionall the vertices of X ( B (cid:48) ) have the same neighborhood in Y . Observe that a vertex x ∈ X ( B (cid:48) ) is connectedto a vertex y iff an odd number of vertices of B (cid:48) are connected to y .For every B (cid:48) ⊆ B , Y ( B (cid:48) ) is the subset of Y containing y if N ( y ) ∩ B = B (cid:48) . It induces a partition of Y intoat most 2 j sets with the same neighborhood in B .Let us finally prove that the partitions of X ( B (cid:48) ) B (cid:48) ⊆B and Y ( B (cid:48) ) B (cid:48) ⊆B satisfy the required properties. Let x, y be in X ( B (cid:48) ) × Y ( B (cid:48)(cid:48) ) such that xy is an edge. Since xy is an edge, an odd number of vertices of B (cid:48) areconnected to y . Since all the vertices of Y ( B (cid:48)(cid:48) ) have the same neighborhood in B , all the vertices of Y ( B (cid:48)(cid:48) )have an odd number of neighbors on B (cid:48) . Thus x is connected to all the vertices of Y ( B (cid:48)(cid:48) ). Since all thevertices of X ( B (cid:48) ) have the same neighborhood in Y , ( X ( B (cid:48) ) , Y ( B (cid:48)(cid:48) )) forms a complete bipartite graph. Lemma 6.
Every ternary tree T with α > labeled leaves has an edge e such that the partition induced by e has at least α/ labeled leaves in both of its two connected components.Proof. Orient every edge of T from the component with less labeled leaves to the other one (when equalityholds, orient arbitrarily). Observe that leaves are sources of this oriented tree. Let v be an internal node of T which is a sink. Consider a component C of T \ v with at least α/ e = vw the edgeof T inducing the partition ( T \ C, C ). Since e is oriented from w to v , the component T \ C has at least α/ e is the edge we are looking for.5 igure 2: The graph G n,(cid:96) of Theorem 8 with n = 4 and (cid:96) = 2. The vertices of the central clique are the vertices of X , theothers are the vertices of Y . Theorem 7.
The distance VC-dimension of a graph with rankwidth k is at most · k +1 + 2 .Proof. Assume by contradiction that the B -hypergraph of a graph G of rankwidth k admits a 2-shatteredset S of size 3(2 k +1 + 1). Let ( T, f ) be a tree decomposition of G achieving rankwidth k . By Lemma 6, thereis an edge e of T such that the partition induced by e has at least 2 k +1 + 1 vertices of S in both connectedcomponents. Let V , V (resp. X, Y ) be the partition of V (resp. S ) induced by e . Let x , . . . , x k +1 +1 and y , . . . , y k +1 +1 be distinct vertices of X and Y respectively.Since S is 2-shattered, for each ( x i , y j ) ∈ X × Y , there is a ball B i,j such that B i,j ∩ S = { x i , y j } where B i,j is chosen with minimum radius. Claim 3.
One of the following holds: • There is an i such that at least k + 1 balls B i,j have their centers in V . • There is a j such that at least k + 1 balls B i,j have their centers in V .Proof. Orient the edges of the complete bipartite graph with vertex set X ∪ Y such that x i → y j if B i,j hasits center in V and x i ← y j otherwise. The average out-degree of the vertices of X ∪ Y is 2 k + . So avertex has out-degree at least 2 k + 1.Assume that the vertex x i ∈ X has out-degree at least 2 k + 1. There exist 2 k + 1 vertices of Y , w.l.o.g. y , . . . , y k +1 , such that x i y , . . . , x i y k +1 are arcs. So the balls B i,j have their centers in V for all j ∈{ , . . . , k + 1 } , and then the first point holds. If a vertex of Y has out-degree at least 2 k + 1, a symmetricargument ensures that the second point holds, which achieves the proof.By Claim 3, we can assume without loss of generality that B (1 , , B (1 , , . . . , B (1 , k + 1) have theircenters in V . We denote by c i and r i respectively the center and the radius of B (1 , i ) and by P i a minimum c i y i -path. By the pigeonhole principle, two P i ’s leave V by the same set of vertices given by the partitionof Lemma 5. Without loss of generality, we assume that these paths are P and P and we denote by z and z respectively their last vertices in V . We finally assume that d ( z , y ) ≤ d ( z , y ). By Lemma 3, theball B ( z , d ( z , y )) is included in B ( c , r ) since z is on a minimum path from c to y . Let z z (cid:48) be thefirst edge of P between z and y (hence z (cid:48) belongs to Y ). By Lemma 5, z (cid:48) is also a neighbor of z since z and z have the same neighborhood in Y . Thus y ∈ B ( z , d ( z , y )). Thus y ∈ B ( z , d ( z , y )) whichcontradicts the hypothesis.Since the rankwidth is equivalent, up to an exponential function, to the cliquewidth [17], Theorem 7implies that every class of graphs with bounded clique-width has bounded distance 2VC-dimension. VC-dimension with bounded distance VC-dimension
Theorem 8.
Let n, (cid:96) be two integers. There exists a graph G n,(cid:96) of distance VC-dimension at most suchthat the VC-dimension of the B (cid:96) -hypergraph of G n,(cid:96) is at least n . roof. The following construction is illustrated on Figure 2. The graph G n,(cid:96) has vertex set X ∪ Y . Theset X contains n vertices denoted by ( x i ) ≤ i ≤ n and Y is a set of (2 (cid:96) − (cid:0) n (cid:1) vertices denoted by y i,jk where1 ≤ k ≤ (cid:96) − ≤ i < j ≤ n . The graph restricted to X is a clique. The graph restricted to Y is adisjoint union of (cid:0) n (cid:1) induced paths on 2 (cid:96) − X ).More formally, for every 1 ≤ i < j ≤ n and k ≤ (cid:96) −
1, the neighbors of the vertex y i,jk are the vertices y i,jk − and y i,jk +1 where y i,j is x i and y i,j (cid:96) is x j . For every i < j , the path x i , y i,j , y i,j , . . . , y i,j (cid:96) − , x j is called the longpath between x i and x j .The 2VC-dimension of the B (cid:96) -hypergraph of G n,(cid:96) is at least n . Indeed the set X is 2-shattered since forevery x i , x j ∈ X , we have B ( y i,j(cid:96) , (cid:96) ) ∩ X = { x i , x j } .The remaining of the proof consists in showing that the distance VC-dimension of G n,(cid:96) is at most 18.Consider an induced subgraph of G n,(cid:96) . The remaining vertices of X are in the same connected componentsince X is a clique. Connected components with no vertices of X form induced paths and then have distanceVC-dimension at most two by Theorem 4. Thus, by Observation 2, Theorem 8 holds if it holds for theconnected component of X . Claim 4.
A shattered set of size at least four has at most two vertices on each long path.Proof.
Let z , z , z be three vertices which appear in this order on the same long path P and z be a vertexwhich is not between z and z on P . By construction, every path between z and z intersects either z or z . So no pair z, p ∈ V × N satisfy B ( z, p ) ∩ X = { z , z } , i.e. { z , z , z , z } is not shattered.Let Z (cid:48) be a shattered set of size at least 19. By Claim 4, we can extract from Z (cid:48) a set Z of size 10 suchthat vertices of Z are in pairwise distinct long paths. For every vertex z i ∈ Z , a nearest neighbor on X is avertex x of X such that d ( x, z i ) is minimum. Each vertex has at most two nearest neighbors which are theendpoints of the long path containing z i .First assume z , z , z in Z have a common nearest neighbor x , i.e. they are on long paths containing x as endpoint. Without loss of generality d ( z , X ) is minimum. Let z, p be such that { z , z } ⊆ B ( z, p ). Since z and z are not in the same long path, free to exchange z and z , a minimum zz -path passes througha vertex y of X . If y = x , then B ( z, p ) contains B ( x, d ( x, z )) by Lemma 3, and then contains z since d ( x, z ) ≥ d ( x, z ). Otherwise up to symmetry y is not an endpoint of the long path containing z . Indeedthe second endpoint of the long path containing z and the second endpoint of the long path containing z are distinct. Otherwise z , z would be in the same long path since there is a unique long path betweenevery pair of vertices of X . Hence a minimum path from y to z is at least d ( z , X ) + 1. In addition aminimum path between y and z has length at most 1 + d ( z , X ). So d ( y, z ) ≥ d ( y, z ). So z is in B ( z, p )and { z , z , z } cannot be shattered.So each vertex of Z has at most two nearest neighbors in X and each vertex of X is the nearest neighborof at most two vertices of Z . Thus every z ∈ Z share a common nearest neighbor with at most two verticesof Z . Since | Z | ≥
10, at least four vertices z , z , z , z of Z have distinct nearest neighbors. Assume w.l.o.g.that d ( z , X ) is minimum.Let z, p ∈ V × N be such that B ( z, p ) contains z , z , z . Let x , x be the endpoints of the longpath containing z (if z ∈ X we consider that x = x = z ). Since nearest neighbors of z , z , z arepairwise disjoint, we can assume w.l.o.g. that the nearest neighbors of z are distinct from x and from x . So a minimum path from z to z passes through x or x and we have d ( x , z ) ≥ d ( x , z ) and d ( x , z ) ≥ d ( x , z ). By Lemma 3, B ( z, p ) also contains z , i.e. Z cannot be not shattered.Note that we did not make any attempt to exactly evaluate the distance VC-dimension of the graph G n,(cid:96) .
4. Erd˝os-P´osa property
Recall that ν (cid:96) and τ (cid:96) respectively denote the packing number and the transversality of the B (cid:96) -hypergraphof G . Chepoi, Estellon and Vax`es proved in [5] that there is a constant c such that for all (cid:96) , every planargraph G of diameter 2 (cid:96) can be covered by c balls of radius (cid:96) . It means that planar graphs of diameter 2 (cid:96) τ (cid:96) ≤ f ( ν (cid:96) ) since two balls of radius (cid:96) necessarily intersect. They conjectured that there exists a linearfunction f such that for every (cid:96) and every planar graph we have τ (cid:96) ≤ f ( ν (cid:96) ). The following result due to Ding,Seymour and Winkler [6] ensures that a polynomial function f exists for any class of graphs of boundeddistance 2VC-dimension. Theorem 9. (Ding, Seymour, Winkler [6]) Each hypergraph of dual VC-dimension d satisfies, τ ≤ · d · ( d + ν + 3) · (cid:18) d + νd (cid:19) Corollary 10.
Let d be an integer. For every graph G ∈ G and every integer (cid:96) , if the distance VC-dimensionof G is at most d , then τ (cid:96) ≤ · d · ( d + ν (cid:96) + 3) · (cid:18) d + ν (cid:96) d (cid:19) Proof.
Let G be a graph. Observation 1 ensures that the B (cid:96) -hypergraph of G is isomorphic to its dualhypergraph. The B (cid:96) -hypergraph of G is a sub-hypergraph (in the sense of hyperedges) of the B -hypergraphof G . Hence the dual 2VC-dimension of the B (cid:96) -hypergraph of G is at most d and then Theorem 9 can beapplied.Theorem 4, 7 and Corollary 10 ensure that B (cid:96) -hypergraphs of K n -minor free graphs and of boundedrankwidth graphs have the Erd˝os-P´osa property. Note that the gap function is a polynomial function whenthe 2VC-dimension is fixed constant. In particular, Corollary 10 implies that every planar graph of diameter2 (cid:96) has a dominating set at distance (cid:96) of size 35200 ( ν (cid:96) = 1, d = 4). Since Theorem 8 ensures that there aresome graphs with bounded distance VC-dimension and unbounded distance 2VC-dimension, Corollary 10raises a natural question. Does the same hold for graphs of bounded distance VC-dimension? The remainingof this section is devoted to answering this question. Theorem 11.
There exists a function f such that, for every (cid:96) , every graph of distance VC-dimension d canbe covered by f ( ν (cid:96) , d ) balls of radius (cid:96) , i.e. τ (cid:96) ≤ f ( ν (cid:96) , d ) . Our proof is based on a result of Matouˇsek linking ( p, q )-property and Erd˝os-P´osa property [16] (Chepoi,Estellon and Vax`es use this method in their paper). Nevertheless our proof is more technically involved sincewe cannot use topological properties as for planar graphs in [5]. A hypergraph has the ( p, q ) -property if forevery set of p hyperedges, q of them have a non-empty intersection, i.e. there is a vertex v in at least q ofthe p hyperedges. The following result, due to Matouˇsek [16], generalizes a result of Alon and Kleitman [2]. Theorem 12. (Matouˇsek [16]) There exists a function f such that every hypergraph H of dual VC-dimension d satisfying the ( p, d + 1) -property satisfies τ ( H ) ≤ f ( p, d )Let d be an integer. Let G be a graph of distance VC-dimension d . By Observation 1, the dual VC-dimension of the B (cid:96) -hypergraph is at most d . Hence if there exists a function p such that, for every (cid:96) andevery graph G of distance VC-dimension d , the B (cid:96) -hypergraph of G satisfies the ( p ( ν (cid:96) , d ) , d + 1)-property,then Theorem 12 will ensures that Theorem 11 holds. So for proving Theorem 11, it suffices to show thatthe size of a set of balls of radius (cid:96) which does not contain ( d + 1) balls intersecting on a same vertex isbounded by a function of ν (cid:96) and d . The remaining of this section is devoted to proving this result. Let A and B be two disjoint sets. An interference matrix M = ( A, B ) is a matrix with | A | rows and | B | columns such that for every ( a, b ) ∈ A × B , the entry m ( a, b ) is a subset of ( A ∪ B ) \{ a, b } . The size of anentry is its number of elements. A k -interference matrix M is an interference matrix which entries have sizeat most k . If A (cid:48) ⊆ A and B (cid:48) ⊆ B , the submatrix M (cid:48) of M induced by A (cid:48) × B (cid:48) is the matrix restricted to theset of rows A (cid:48) and the set of columns B (cid:48) which entries are m (cid:48) ( a (cid:48) , b (cid:48) ) = m ( a (cid:48) , b (cid:48) ) ∩ ( A (cid:48) ∪ B (cid:48) ). A 0-interferencematrix is called a proper matrix . A matrix is square if | A | = | B | . The size of a square matrix is its numberof rows. 8 emma 13. Let k > . A k -interference square matrix with no proper submatrix of size n has size lessthan kn .Proof. Let us show that if M = ( A, B ) is a k -interference matrix with size m = kn + 1, then it contains aproper submatrix of size n . A triple ( i, j, l ) ∈ A × B × ( A ∪ B ) is a bad triple if l ∈ m ( i, j ) (and then l (cid:54) = i and l (cid:54) = j ). A bad triple ( i, j, l ) is bad for ( X, Y ) with X ⊆ A , Y ⊆ B , | X | = | Y | = n if i ∈ A, j ∈ B and l isin A or B .For a given bad triple ( i, j, l ), let us count the number of pairs ( X, Y ) where X ⊆ A , Y ⊆ B , and | X | = | Y | = n containing ( i, j, l ) as a bad triple. Let us consider the case l ∈ A (the case l ∈ B is obtainedsimilarly). The number of X ’s containing both i and l is (cid:0) m − n − (cid:1) since i (cid:54) = l . The number of Y ’s containing j is (cid:0) m − n − (cid:1) . Since M is a k -interference matrix, the total number of bad triples is at most k · m . Thus thetotal number of pairs X, Y with X ⊆ A , Y ⊆ B , | X | = | Y | = n is (cid:0) nm (cid:1) . So if the number of such pairs islarger than the number of pairs containing a bad triple, the conclusion holds. In other words, if (cid:18) m − n − (cid:19) · (cid:18) m − n − (cid:19) · km < (cid:18) nm (cid:19) there is a pair ( X, Y ) with X ⊆ A , Y ⊆ B , | X | = | Y | = n which does not contain a bad triple. This latterinequality is equivalent with kn · ( n − < m − P from x to y and a path Q from y to z , the concatenation of P and Q denoted by P Q isthe walk consisting on the edges of P followed by the edges of Q . The length of a path P is denoted by | P | .Let G = ( V, E ) be a graph and ≺ l be a total order on E . We extend ≺ l on paths, for any paths P and P as follows : • If P has no edges, then P ≺ l P . • If P = P (cid:48) .e and P = P (cid:48) .e , where e is the last edge of P and P , then P ≺ l P if and only if P (cid:48) ≺ l P (cid:48) . • If P = P (cid:48) .e and P = P (cid:48) .e , where e (cid:54) = e , then P ≺ l P if and only if e ≺ l e .The order ≺ l is called the lexicographic order (note nevertheless that paths are compared from their endto their beginning). The minimum path from x to z , also called the xz -path and denoted by P xz , is the pathof minimum length with minimum lexicographic order from x to z . Observe that two minimum paths goingto the same vertex z and passing through the same vertex u coincide between u and z . We note u (cid:69) xz v if u appears before v on the xz -path. Given a path from a to b passing through c , the suffix path on c (resp. prefix path on c ) is the cb -subpath (resp. ac -subpath) of the ab -path. Note that every suffix of a minimumpath is a minimum path. Given two sets X and Z , the XZ -paths are the xz -paths for all x, z ∈ X × Z .Let x , x and z be three vertices. Two distinct edges v u and u v form a cross between the x z -pathand the x z -path if for i ∈ { , } , u i (cid:69) x i z v i (see Figure 3). Lemma 14.
Let x , x , z be three vertices. If the edges u v and v u form a cross between the x z -pathand the x z -path, then free to exchange x and x we have: • either u = v and u v is an edge. • Or u = v and the v z -path is the edge v v concatenated with the v z -path.In other words, only cases (c) and (d) of Figure 3 can occur.Proof. For i ∈ { , } , we denote by Q u i (resp. Q v i ) the suffix of the x i z -path on u i (resp. v i ). Since suffixesof minimum paths are minimum paths, these four paths are minimum paths. We prove that if a cross doesnot satisfy the condition of Lemma 14, then one of these paths is not minimum.9 x u v u v zd d z u = v u v ( a ) ( b ) ( c ) d − dx x z u = v v u x x z u = v u v ( d ) d − x x Figure 3: 4 types of crosses. Lemma 14 ensures that, up to symmetry, only (c) and (d) are authorized. The thick chords areedges of the graph. Thin chords represent paths. Distances are denoted by d or d −
1. In the case of Figure 3(d), the path Q v is v v Q v . A Baa bb s ab s a b s a b s ab Figure 4: The sets S ab = { s ab } are 2-disconnecting for A, B . A real cross is a cross for which u (cid:54) = v and u (cid:54) = v (Figure 3(a)). A degenerated cross is a cross forwhich, up to symmetry, u = v and Q v (cid:54) = v v .Q v (Figure 3(b)).A real cross satisfies | Q v | = | Q v | . Indeed if | Q v | < | Q v | then u v .Q v has length at most | Q v | . Thispath is strictly shorter than Q u (since u (cid:54) = v , indeed the cross is a real cross), contradicting the minimalityof Q u . So | Q v | = | Q v | . Free to exchange x and x , we have Q v ≺ l Q v . So u v .Q v ≺ l Q u (recall thatwe first compare the last edge) and | u v .Q v | ≤ | Q u | . So Q u is not minimum, a contradiction. Hencethere is no real cross.Consider a degenerated cross such that u v / ∈ E . In particular u and v are at distance 2. So we have | Q v | < | Q v | otherwise u v .Q v would be strictly shorter than Q u , a contradiction. In addition, | Q v | and | Q v | differ by at most one since v v is an edge. So | Q v | + 1 = | Q v | . Assume now that we are not in thecase of Figure 3(d), in other words, Q v (cid:54) = v v Q v . If Q v ≺ l Q v then u v Q v is not longer than Q u (since u and v are at distance 2) and has a smaller lexicographic order, a contradiction with the minimalityof Q u . If Q v ≺ l Q v then v v .Q v is not longer and has a smaller lexicographic order, a contradictionwith the minimality of Q v . So either the degenerated cross satisfies u v ∈ E or Q v = v v .Q v .Let (cid:96) be an integer and A, B be two disjoint subsets of vertices. To every pair ( a, b ) ∈ A × B , we associatea set of vertices S a,b which is disjoint from A ∪ B . We say that the set of subsets S = { ( S a,b ) ( a,b ) ∈ A × B } is (cid:96) -disconnecting if for every subset C of S and every pair ( a, b ), we have d ( a, b ) > (cid:96) in G \ (cid:83) C if and only if S a,b ∈ C . If such a family of sets exists, then A, B are said to be (cid:96) -disconnectable . Another way of defining (cid:96) -disconnecting families would be to say that d ( a, b ) > (cid:96) in G \ S a,b and d ( a, b ) ≤ (cid:96) in G \ (cid:83) ( S \ S a,b ), orroughly speaking that S a,b is the only set whose deletion can increase d ( a, b ) above (cid:96) . In Figure 4, the sets A, B are 2-disconnectable. Indeed the deletion of any vertex s ab eliminates all the paths of length at most 2from a to b . Note nevertheless that a and b are still in the same connected component after this operation. Theorem 15.
Let G = ( V, E ) be a graph and (cid:96) be an integer. If there exist two subsets A, B of V with | B | = 2 | A | which are (cid:96) -disconnectable, then the distance VC-dimension of G is at least | A | . roof. Let us prove that the set A can be shattered in the B (cid:96) -hypergraph of an induced subgraph of G .Associate in a one to one way every vertex b of B to a subset A b of A . Since A, B are (cid:96) -disconnectable, thereexists a family S of subsets which is (cid:96) -disconnecting for A, B . Let C be the collection of S consisting of allthe sets S a,b such that a ∈ A b . Since S is (cid:96) -disconnecting, B ( b, (cid:96) ) ∩ A = A \ A b in G \ C , for all b ∈ B (thedeletion of S a,b eliminates the paths of length at most (cid:96) between a and b ). Hence the set A is shattered byballs of radius (cid:96) in G \ C . Therefore the distance VC-dimension of G is at least | A | . Let G be a graph of distance VC-dimension d and q, (cid:96) be two integers. Most of the following definitionsdepend on (cid:96) . Nevertheless, in order to avoid heavy notations, this dependence will be implicit in theterminology. A set of balls of radius (cid:96) is q -sparse if no vertex of the graph is in more than q balls of theset. Note that a subset of a q -sparse set is still q -sparse. By abuse of notation, a set X of vertices is called q -sparse if the set of balls of radius (cid:96) centered in X is q -sparse.Assume that the B (cid:96) -hypergraph of a graph G does not satisfy the ( p, d + 1)-property. Then there exist p balls of radius (cid:96) such that no vertex is in at least ( d + 1) of these p balls, i.e. there is a d -sparse set of size p .In other words, a d -sparse set of size p is a certificate that the ( p, d + 1)-property does not hold. In order toprove Theorem 11, we just have to show that p can be bounded by a function of d and ν l . The remaining ofthis section is devoted to show that there exists a function f such that the size of a d -sparse set is at most f ( d, ν l ).A set X of vertices is d -localized if the vertices of X are pairwise at distance at least (cid:96) + 1 and at most2 (cid:96) − d +2 −
3. A d -localized set is defined only if this value is positive. A pair A, B of disjoint sets of verticesis q -sparse if A ∪ B is. A disjoint pair A, B of vertices is d -localized if the vertices of A ∪ B are pairwise atdistance at least (cid:96) + 1, and if for every a, b ∈ A × B , d ( a, b ) ≤ (cid:96) − d +2 −
3. A subpair of a d -localized pairis d -localized. The size of a pair A, B is min( | A | , | B | ). Theorem 16. (Ramsey) There exists a function r k such that every complete edge-colored graph G with k colors with no monochromatic clique of size n has at most r k ( n ) vertices. All along the paper, logarithms are in base 2.
Theorem 17.
Let G be a graph and X be a subset of vertices pairwise at distance exactly r . Assume alsothat no vertex of G belongs to q balls of radius (cid:100) r/ (cid:101) with centers in X . Then the distance VC-dimension of G is at least (log | X | − log 2 q ) / .Proof. Let r (cid:48) be equal to (cid:100) r/ (cid:101) . Free to remove one vertex from X , we can assume that X is even, andwe consider a partition A, B of X with | A | = | B | . For every pair ( a, b ) ∈ A × B , we denote the minimum ab -path by P ab . By abuse of notation, we still denote by G the restriction of G to the vertices of the unionof the paths P ab for all a ∈ A and b ∈ B . Observe that we preserve the hypothesis of Theorem 17 apartfrom the fact that the distance between vertices inside A (resp. inside B ) may have increased above r . Let a ∈ A and b ∈ B . Let y be a vertex of X distinct from a and b . If B ( y, r (cid:48) ) ∩ P ab (cid:54) = ∅ , then denote by x a vertex in this set. We have d ( a, x ) ≥ (cid:98) r/ (cid:99) since d ( a, y ) ≥ r and d ( y, x ) ≤ (cid:100) r/ (cid:101) . By symmetry, we alsohave d ( b, x ) ≥ (cid:98) r/ (cid:99) . Hence x is a midvertex of P ab , i.e. a vertex of P ab at distance (cid:98) r/ (cid:99) or (cid:100) r/ (cid:101) from a (and thus also from b ). Recall that a midvertex x of P ab belongs to at most q − r (cid:48) (including B ( a, r (cid:48) ) and B ( b, r (cid:48) )).Consider the interference matrix M = ( A, B ) where m ( a, b ) = { y ∈ ( A ∪ B ) \{ a, b }| B ( y, r (cid:48) ) ∩ P ab (cid:54) = ∅ } .Since P ab has at most two midvertices and each of these belongs to at most q − B ( y, r (cid:48) ) with y different from a and b , the matrix M is a (2 q − M is a 2 q -interference matrix (with 2 q ≥ M (cid:48) of size N = ( | X | / q ) / . Let us denote by A (cid:48) the set of rowsand B (cid:48) the set of columns of the extracted matrix. Let us still denote by G the restriction of the graph tothe vertices of the paths ( P ab ) ( a,b ) ∈ A (cid:48) × B (cid:48) .Let a, a (cid:48) ∈ A (cid:48) and b (cid:48) ∈ B (cid:48) . The key-observation is that if B ( a, r (cid:48) ) intersects P a (cid:48) b (cid:48) , then a = a (cid:48) . Indeed,by definition of M , we have a ∈ m ( a (cid:48) , b ), contradicting the fact that M (cid:48) is a proper submatrix.11et M ab be the set of midvertices of P ab , where a, b ∈ A (cid:48) × B (cid:48) . We claim that M ab is disjoint from P a (cid:48) b (cid:48) ,whenever P a (cid:48) b (cid:48) (cid:54) = P ab . Indeed if x ∈ M ab ∩ P a (cid:48) b (cid:48) , we have in particular both d ( a, x ) ≤ r (cid:48) and d ( b, x ) ≤ r (cid:48) , andthus by the key-observation a = a (cid:48) and b = b (cid:48) . In other words, deleting M ab never affects P a (cid:48) b (cid:48) , whenever P a (cid:48) b (cid:48) (cid:54) = P ab .Another crucial remark is that every path P of length r from a to b intersects M ab . Indeed, let x be avertex of P with both d ( a, x ) ≤ r (cid:48) and d ( b, x ) ≤ r (cid:48) . Since x is in G , it belongs to some path P a (cid:48) b (cid:48) . By thekey-observation, we both have a (cid:48) = a and b (cid:48) = b , hence x ∈ M ab .To conclude, observe that the deletion of M ab ensures that the distance d ( a, b ) is more than r whereasdeleting the union of all M a (cid:48) b (cid:48) different from M ab does not affect d ( a, b ) which is still equal to r . Consequently,the sets ( M ab ) ( a,b ) ∈ A (cid:48) × B (cid:48) are r -disconnecting for A (cid:48) , B (cid:48) . Hence, by Theorem 15, the distance VC-dimensionof G is at least log( N ) = (log | X | − log 2 q ) / Lemma 18.
Let G be a graph of distance VC-dimension at most d . There exists a function f such that:(a) Either G contains a d -localized set of size p which is d -sparse,(b) Or the ( f ( ν (cid:96) , d, p ) , d + 1) -property holds.Proof. Let D = 2 d +2 + 2 and N = max( p, ν l + 1 , d +3+log(4 d +2) ). Let f be a function such that f ( ν (cid:96) , d, p ) ≥ r D +4 ( N ) + 1. Let us show that function f satisfies Lemma 18. Assume that point (b) does not hold, i.e. the ( f ( ν (cid:96) , d, p ) , d + 1)-property does not hold. So there is a subset X of vertices of size r D +4 ( N ) + 1 suchthat the set X is d -sparse. Let us show that point (a) holds.Consider the complete ( D + 4)-edge-colored graph G (cid:48) with vertex set X such that, for every x, y ∈ X , xy has color: • c with 0 ≤ c ≤ D if d ( x, y ) = 2 (cid:96) − c , • D + 1 if d ( x, y ) ≤ (cid:96) , • D + 2 if d ( x, y ) > (cid:96) , • D + 3 otherwise.Theorem 16 ensures that there is a monochromatic clique K of size N . Let K (cid:48) be a clique of color D + 1and x ∈ K (cid:48) . Then K (cid:48) ⊆ B ( x, (cid:96) ) ∩ X . Thus the size of K (cid:48) is at most d since X is d -sparse. At most ν (cid:96) balls of radius (cid:96) centered in X are vertex disjoint by definition of the packing number. Thus the size of aclique of color D + 2 is at most ν (cid:96) < N . Since X is d -sparse, then K also is. Then, for every 0 ≤ c ≤ D ,no vertex of G belongs to ( d + 1) balls of radius (cid:100) (2 (cid:96) − c ) / (cid:101) ≤ (cid:96) centered in X . Therefore the color of K cannot be in 0 ≤ r ≤ D . Otherwise Theorem 17 would ensure that the distance VC-dimension of G is atleast log( N ) / − log(4 d + 2) / ≥ d + 1. So the clique K of size N ≥ p has color D + 3. A clique of color D + 3 defines a d -localized set. Moreover K is d -sparse since X is. Thus K satisfies (a).The vertices of a d -localized set have to be pairwise at distance at least d + 1 and at most 2 (cid:96) − d +2 − In this section we introduce a notion of independence for every pair of vertices. We first give someproperties of independent pairs and we will finally show that any large enough d -sparse and d -localized paircontains a large enough independent subpair.Let A, B be a d -localized pair. In the following we consider the restriction of the graph to ∪ a ∈ A,b ∈ B P ab .Recall that P ab is the minimum path with minimum lexicographic order from a to b , also called the ab -path.Note that the sets A and B are not treated symmetrically since we only consider the minimum paths from A to B . Let a, a (cid:48) ∈ A and b ∈ B . Note that, since d ( a, a (cid:48) ) > (cid:96) , d ( a (cid:48) , b ) > (cid:96) and d ( a, b ) < (cid:96) , the vertex a (cid:48) does not belong to P ab . 12 a b b RS ( a ) c ab c ab c − ab c − ab BARS ( a ) Figure 5: Minimum paths with root sections (dashed parts), critical vertices and pre-critical vertices.
For every pair a, b ∈ A × B , the critical vertex c ab (resp. c ba ) is the vertex of P ab at distance (cid:96) − a (resp. b ) and the pre-critical vertex c − ab is the vertex of P ab at distance (cid:96) − a (see Figure 5). Suchvertices exist since d ( a, b ) > (cid:96) . Moreover c ab and c − ab are adjacent. Note that both c ab and c ba are verticesof P ab . In the following, we mostly need the vertex c ba in order to ensure some distance properties (andthen we do not use the minimality of the lexicographic order for these vertices). On the contrary, the vertex c ab will be used for both distance and lexicographic arguments. The root section of a ∈ A (resp. b ∈ B ),denoted by RS ( a ) (resp. RS ( b )), is the set of vertices of the ac ab -subpaths (resp. c ba b -subpaths) of P ab forall b ∈ B (resp. a ∈ A ). We denote by RS ( A ) the set ∪ a ∈ A RS ( a ).Since d ( a, b ) ≤ (cid:96) −
7, the vertex c ba precedes the vertex c ab on the path P ab . In particular we have P ab ⊆ RS ( a ) ∪ RS ( b ), hence every vertex of G belongs to some root section. In fact, we have the slightlystronger following observation: Observation 19.
For every a, b in A × B , the critical vertex c ab and the pre-critical vertex c − ab are in RS ( b ) . A d -localized pair A, B is independent , if for every a, b ∈ A × B , the ball B ( c ab , (cid:96) ) intersects A ∪ B on { a, b } and B ( c ba , (cid:96) ) ∩ ( A ∪ B ) = { a, b } . A subpair of an independent pair is still independent. In addition, A, B is still independent in the graph induced by the vertices of the AB -paths. Lemma 20.
The size of a d -sparse and d -localized pair with no independent subpair of size p is at most d · p .Proof. Let
A, B be a d -sparse and d -localized pair of size 2 d · p + 1. For every vertex u , I ( u ) denotes B ( u, (cid:96) ) ∩ ( A ∪ B ). Since A ∩ B = ∅ , the matrix M = ( A, B ) where m ( a, b ) = ( I ( c ab ) ∪ I ( c ba )) \{ a, b } , is awell-defined interference matrix. The pair A, B is d -sparse, then | I ( u ) | ≤ d for every vertex u . Thus M is a2 d -interference matrix.By Lemma 13, M has a proper submatrix ( A (cid:48) , B (cid:48) ) of size p . Thus for every a (cid:48) , b (cid:48) ∈ A (cid:48) × B (cid:48) , B ( c a (cid:48) b (cid:48) , (cid:96) ) ∩ ( A (cid:48) ∪ B (cid:48) ) = { a (cid:48) , b (cid:48) } and the same holds for c b (cid:48) a (cid:48) , i.e. A (cid:48) , B (cid:48) is independent. Lemma 21.
Let
A, B be an independent pair.(a) Every pair of vertices of endpoints disjoint AB -paths are at distance at least .(b) For every pair a, a (cid:48) in A (resp. b, b (cid:48) in B ), d ( RS ( a ) , RS ( a (cid:48) )) ≥ (resp. d ( RS ( b ) , RS ( b (cid:48) ) ≥ ).Proof. Let us first prove (b). We prove it for vertices of A , the case of vertices of B will handle symmetrically(indeed the proof rely on distance arguments and not lexicographic ones). Let a (cid:54) = a (cid:48) with u ∈ RS ( a ) and u (cid:48) ∈ RS ( a (cid:48) ). There exists b and b (cid:48) in B such that u is in the prefix path on c ab of the ab -path and u (cid:48) is in theprefix path on c a (cid:48) b (cid:48) of the a (cid:48) b (cid:48) -path. Free to exchange a and a (cid:48) , d ( a, u ) ≤ d ( a (cid:48) , u (cid:48) ). Since d ( a (cid:48) , c a (cid:48) b (cid:48) ) = (cid:96) − a bc ab c a b u v a a uc ab c a b v ba a bu = c ab c a b v Figure 6: Examples of escapes. The right one is an edge of the a (cid:48) b -path. we have d ( a, u ) + d ( u (cid:48) , c a (cid:48) b (cid:48) ) ≤ d ( a (cid:48) , u (cid:48) ) + d ( u (cid:48) , c a (cid:48) b (cid:48) ) = (cid:96) −
3. Since
A, B is independent, d ( a, c a (cid:48) b (cid:48) ) > (cid:96) , wehave (cid:96) < d ( a, u ) + d ( u, u (cid:48) ) + d ( u (cid:48) , c a (cid:48) b (cid:48) ) ≤ (cid:96) − d ( u, u (cid:48) ), and then d ( u, u (cid:48) ) ≥
4. So (b) holds.Let u be a vertex of the ab -path, and u (cid:48) be a vertex of the a (cid:48) b (cid:48) -path such that a (cid:54) = a (cid:48) and b (cid:54) = b (cid:48) .By part (b) of Lemma 21 we may assume without loss of generality that u ∈ RS ( a ) and u (cid:48) ∈ RS ( b (cid:48) ). Inaddition, we can assume that d ( a, u ) ≤ d ( b (cid:48) , u (cid:48) ). So d ( a, u ) + d ( u (cid:48) , c b (cid:48) a (cid:48) ) ≤ d ( b (cid:48) , u (cid:48) ) + d ( u (cid:48) , c b (cid:48) a (cid:48) ) = (cid:96) −
3. So (cid:96) < d ( a, c a (cid:48) b (cid:48) ) ≤ d ( a, u ) + d ( u, u (cid:48) ) + d ( u (cid:48) , c b (cid:48) a (cid:48) ) ≤ (cid:96) − d ( u, u (cid:48) ). Hence d ( u, u (cid:48) ) ≥ leaves a set S if exactly one of its endpoints is in S . Observation 22.
Let
A, B be an independent pair and a ∈ A . For all b (cid:54) = b (cid:48) , we have c − ab (cid:54) = c − ab (cid:48) (and then c ab (cid:54) = c ab (cid:48) ). Moreover the edges of the aB -paths leaving RS ( a ) form an induced matching. Recall that, by lexicographic minimality, when two aB -paths separate, they never meet again, so if c − ab (cid:54) = c − ab (cid:48) , we immediately have c ab (cid:54) = c ab (cid:48) . Proof.
Observation 19 ensures that c − ab ∈ RS ( b ) and c − ab (cid:48) ∈ RS ( b (cid:48) ). So Lemma 21(b) ensures that c − ab (cid:54) = c − ab (cid:48) .The lexicographic minimality ensures that edges of aB -paths leaving RS ( a ) are vertex disjoint, i.e. theyform a (non necessarily induced) matching. By Observation 19, the edge of P ab leaving RS ( a ) is an edgewith both endpoints in RS ( b ). Thus Lemma 21(b) ensures that the matching is induced. Let
A, B be an independent pair. In the following we consider the restriction of the graph to the verticesof the AB -paths. Let a in A . An escape uv from a is an edge leaving RS ( a ) such that uv is not an edge ofany P ab for b ∈ B . By convention, when uv is an escape from a , we still denote by u the vertex in RS ( a )and by v the vertex which is not in RS ( a ). The vertex u is called the beginning of the escape and v the end of the escape.Let uv be an escape from a . Since u ∈ RS ( a ), there exists b ∈ B such that the vertex u is in P ab . Sincewe have considered the restriction of the graph to the vertices of the AB -paths, the vertex v is in the path P a (cid:48) b (cid:48) for a (cid:48) ∈ A and b (cid:48) ∈ B . Lemma 21(a) ensures that either a = a (cid:48) or b = b (cid:48) . If a = a (cid:48) then d ( a, v ) > (cid:96) − v would be in RS ( a )). So we have d ( a, u ) = (cid:96) − u ∈ RS ( a ) and uv is an edge. Thoughthe induced matching property of Observation 22 ensures that there is no edge between c ab and v (otherwisethe edges leaving P ab and P ab (cid:48) do not form an induced matching). So a (cid:54) = a (cid:48) , i.e. b = b (cid:48) . Thus every escape uv an escape from a to a (cid:48) for b . In Figure 6, the edges uv are escapes from a to a (cid:48) for b . An escape can bean edge of a minimum path (see the rightmost example of Figure 6).A deep escape is an escape such that u is neither a critical vertex nor a pre-critical vertex. Let us definetwo graphs: the escape graph of b (resp. deep escape graph of b ) is a directed graph with vertex set A where aa (cid:48) is an arc if there is an escape (resp. a deep escape) from a to a (cid:48) for b . In Figure 6, the leftmost escapeis not a deep escape since u = c ab .If a vertex x which is not in RS ( A ) has a neighbor in RS ( a ), a is called an origin root section on x .Lemma 21(b) ensures that every vertex has at most one origin root section (otherwise two root sectionswould be at distance 2). Note that if uv is an escape from a , then a is the origin root section of v .14et us informally explain why we introduce escapes. As long as a path from a to B follow edges of aB -paths, then we can understand the structure of the path. In particular, if such a path passes through acritical (or pre-critical vertex) we can “evaluate” its length using the fact that d ( a, c ab ) = (cid:96) −
3. If a pathuses an escape, it can “escape” RS ( a ) without passing through such a vertex, which implies that the lengthof the path is somehow harder to evaluate. Let us first show that the structure of the (deep) escape graphcan be constraint. Lemma 23.
Let
A, B be an independent pair. For every b ∈ B , the escape graph of b has no circuit.Proof. Assume that there is a circuit a , a , . . . , a k , a . In the following indices have to be understood modulo k + 1. For every i , let u i v i be an escape from a i to a i +1 for b . Since u i ∈ RS ( a i ) and u i +1 ∈ RS ( a i +1 ),Lemma 21(b) ensures that d ( u i , u i +1 ) ≥
4, then d ( v i , u i +1 ) ≥
3. Hence d ( b, u i ) ≤ d ( b, v i ) + 1 < d ( b, v i ) + d ( v i , u i +1 ) = d ( b, u i +1 ). The first inequality comes from the fact that u i v i is an edge and the last equalitycomes from the fact that the path is a minimum path. A propagation of these inequalities along the arcs ofthe circuit leads to d ( b, u ) < d ( b, u ), a contradiction.The deep escape graph of b is a subgraph, in the sense of arcs, of the escape graph of b . Thus the deepescape graph of b has no circuit. For every b , the order inherited from b is a partial order on A such that a < a (cid:48) if and only if there is an escape from a to a (cid:48) for b . An independent pair A, B has the escape property if for every b ∈ B , the deep escape graph of b is a transitive tournament. Lemma 24.
The size of an independent pair with no subpair of size d +1 satisfying the escape property isat most r d +2 (2 d +1 ) .Proof. Let (
A, B ) be an independent pair of size r d +2 (2 d +1 ) + 1. Claim 5.
A, B has a subpair
X, Z of size d +1 such that:(1) either the pair X, Z does not contain a deep escape,(2) or the pair
X, Z satisfies the escape property.Proof.
Let B (cid:48) = { b , . . . , b d +2 } be a subset of B of size 2 d +2 . Consider the complete edge-colored graph G (cid:48) on vertex set A . The colors are binary integers of 2 d +2 digits. The i -th digit of the color of aa (cid:48) is 1 if thereis a deep escape from a to a (cid:48) (or from a (cid:48) to a ) for b i and 0 otherwise. Theorem 16 ensures that G (cid:48) containsa monochromatic clique X of size 2 d +1 . Let us denote by c the color of the edges of G (cid:48) [ X ]. At least 2 d +1 digits of c are equal. Denote by Z the subset of B (cid:48) corresponding to these digits. If the digits equal 0 then(1) holds, otherwise (2) holds.Let us prove by contradiction that Claim 5(1) cannot hold. Let X, Z be an independent pair with nodeep escape. Consider the restriction of the graph to (cid:83) x,z P xz . For every x, z , the private part of xz , denotedby P P ( x, z ), is the set of vertices which belong to P xz and which do not belong to any other path in P XZ . Claim 6.
P P ( x, z ) separates x from c xz and from c − xz in the graph induced by RS ( x ) .Proof. Let P be a path from x to c xz in RS ( x ) and let u be the last vertex of P which is on P xz (cid:48) for z (cid:48) (cid:54) = z .The vertex u exists since c xz (cid:54) = c xz (cid:48) and x ∈ P xz (cid:48) for every z (cid:48) (cid:54) = z . Let v be the vertex after u in P . Bymaximality of u , the vertex v is in P xz (since v ∈ RS ( x )). So if v / ∈ P P ( x, z ) then v ∈ P x (cid:48) z (cid:48)(cid:48) for some x (cid:48) (cid:54) = x .By Lemma 21(a), we have z = z (cid:48)(cid:48) . Thus a vertex of P xz (cid:48) and a vertex of P x (cid:48) z are adjacent, contradictingLemma 21(a).Let P be a path from x to c − xz which does not pass through P P ( x, z ). Since P c − xz c xz is a path from x to c xz , the first part of the proof ensures that c xz ∈ P P ( x, z ). Since c − xz / ∈ P P ( x, z ), the lexicographicminimality ensures that c − xz is in P x (cid:48) z (cid:48) for z (cid:54) = z (cid:48) . Lemma 21(a) ensures that x = x (cid:48) . By Observation 19, wehave c − xz ∈ RS ( z (cid:48) ) and c xz ∈ RS ( z ), contradicting Lemma 21(b).15 a bc ab c a b v a b c a b a Figure 7: The vertex v is the incoming vertex of the a (cid:48) b -path. The gray part (where v a (cid:48) b is included but not c a (cid:48) b ) is the freesection of the a (cid:48) b -path. Let us finally prove that
X, Z is (2 (cid:96) − P P ( x, z ). Let x, z ∈ X, Z . Since
X, Z is d -localized, P xz has length at most 2 (cid:96) −
7. In addition
P P ( x, z ) does not intersect P x (cid:48) z (cid:48) if x (cid:54) = x (cid:48) or z (cid:54) = z (cid:48) ; so the deletion of P P ( x, z ) does not delete all the paths from x (cid:48) z (cid:48) of length at most 2 (cid:96) −
7. Let usfinally show that all the paths of length at most 2 (cid:96) − x to z pass through P P ( x, z ).Since there is no deep escape, any edge leaving RS ( x ) intersects a critical or a pre-critical vertex. Byindependence, if z (cid:54) = z (cid:48) then we have d ( c xz (cid:48) , z ) ≥ (cid:96) +1 and d ( c − xz (cid:48) , z ) ≥ (cid:96) . Moreover, we have d ( x, c xz (cid:48) ) = (cid:96) − d ( x, c − xz (cid:48) ) = (cid:96) −
4. Thus the length a path from x to z passing through c xz (cid:48) or c − xz (cid:48) is at least 2 (cid:96) − (cid:96) − x to z passes through c xz or c − xz . By Claim 6, there isno path of length at most 2 (cid:96) − x to z in G [ V \ P P ( x, z )]. By Theorem 15, the distance VC-dimensionis at least ( d + 1), a contradiction. So case (1) of Claim 5 cannot hold, i.e. case (2) holds. The outline of the proof of Lemma 24 consisted in finding a (2 (cid:96) − a (cid:48) , b ∈ A × B . The incoming vertex v a (cid:48) b of the a (cid:48) b -path is the first vertex in P a (cid:48) b (from a (cid:48) to b ) for which there exists an escape u a (cid:48) b v a (cid:48) b from a to a (cid:48) for b for some a ∈ A . In other words, it is the first vertex of P a (cid:48) b at distance one from RS ( A ) \ RS ( a (cid:48) ).The edge u a (cid:48) b v a (cid:48) b is a first-in escape to a (cid:48) for b . Note that several first-in escapes to a (cid:48) can exist, but theincoming vertex is unique. The free section of the a (cid:48) b -path, denoted by F S ( a (cid:48) , b ), is the c a (cid:48) b v a (cid:48) b -subpath ofthe a (cid:48) b -path where c a (cid:48) b is not included but v a (cid:48) b is included. Lemma 21(b) ensures that the free section existsand has length at least 3. Lemma 25.
Let
A, B be a pair satisfying the escape property. Then there is no edge between two free sectionsof AB -paths.Proof. Consider an edge xy where x ∈ F S ( a (cid:48) , b (cid:48) ) and y ∈ F S ( a, b ). Let us prove that there is a forbiddencross (see Lemma 14). Notice that x ∈ RS ( b (cid:48) ) since x ∈ P a (cid:48) b (cid:48) ( F S ( a (cid:48) , b (cid:48) ) is a subpath of P a (cid:48) b (cid:48) ) and x / ∈ RS ( a (cid:48) )(it is after c a (cid:48) b (cid:48) ). Similarly, y ∈ RS ( b ). So Lemma 21(b) ensures that b = b (cid:48) . Assume w.l.o.g. that a < a (cid:48) in the order inherited from b . Hence there is a deep escape uv from a to a (cid:48) for b . By definition of deepescape, y is strictly after c ab in P ab and u is strictly before c − ab in P ab . So we have d ( u, y ) ≥ P ab isa minimum path). Moreover x is before v on P a (cid:48) b by definition of the free section of P a (cid:48) b . Finally edges xy and uv contradict Lemma 14. Lemma 26.
The size of a pair with the escape property is at most d +1 − .Proof. Assume by contradiction that a pair
A, B of size 2 d +1 satisfies the escape property. Let b ∈ B . Letus denote by a , . . . , a d +1 the vertices of A ordered along the order inherited from b . For every i ≥
2, wedenote by v i the incoming vertex and by u i v i a first-in escape to a i for b . By convention we put v = b and F S ( a , b ) is the subpath of P a b from c a b to b . Recall that there exists j < i such that u i ∈ RS ( a j ).Note that v j is after u i on P a j b . Indeed u i appears before c a j b since u i ∈ RS ( a j ) and v j appears after c a j b .Therefore the following collection of Ab -paths, called jump paths (for b ), is well-defined:16 vu v Figure 8: The minimum path (at the left) is transformed into the jump path (at the right). • The jump path of a b is the a b -path. • The jump path of a i b is the a i v i -subpath of P a i b , the edge v i u i of origin root section a j and the suffixpath on u i of the jump path of a j b (see Figure 8).Note that jump paths can be equal to minimum paths (see the rightmost part of Figure 6).Let us analyze a bit the structure of jump paths. The jump path of a i b starts with the a i v i -subpath of P a i b . In particular both paths coincide in RS ( a i ). Then the jump path of a i b contains the first in-escapeto a i , namely the edge v i u i . By definition of the order, the vertex u i is in RS ( a j ) for j < i and then u i v i is an escape from a j to a i for b . Thus u i is in P a j b ∩ RS ( a j ). So it is on the jump path of a j b . After this“rerouting” the two jump paths are the same and do not quit each other before the end of the path.Jump paths follow minimum AB -paths except on incoming vertices in which they are “rerouted”. A rerouting edge is an edge e such that there exists i satisfying e = u i v i . Since after a rerouting edge, the jumppath of a i b coincides with the jump path a j b for j < i , every jump path has at most 2 d +1 rerouting edges.Moreover each rerouting edge increases the length of the path by at most two since | d ( u i , b ) − d ( v i , b ) | ≤ u i v i is an edge and P a j b is minimum). Since a pair with the escape property is d -localized (each path P ab has length at most 2 (cid:96) − d +2 − ab is at most(2 (cid:96) − d +2 −
3) + 2 d +1 · (cid:96) − a, b . Let us now state a claim on the structure of the paths. Claim 7.
Any vertex of a jump path is either in RS ( A ) or in a free section F S ( a, b ) . Moreover any vertexof a jump path for b is in (cid:83) a ∈ A P ab .Proof. By induction on the order inherited from b . It holds for the jump path of a b . The jump path of a i b coincides with the a i b -path from a to the incoming vertex, i.e. on RS ( a i ) and on F S ( a i , b ). By induction,it holds for the remaining vertices since the remaining of the jump path of a i b is included in the jump pathof a j b for j < i .In the remaining of the proof we consider the restriction of the graph to the vertices of the jump pathsof ab for every a, b ∈ A × B . Let a i ∈ A, b ∈ B . Remind that the first vertex of F S ( a i , b ) is the vertex after c a i b in P a i b and the last one is v i , the incoming vertex of P a i b . Claim 8.
Let i ≥ . The vertices of F S ( a i , b ) induce a subpath w , . . . , w k = v i of P a i b . The only neighborsof these vertices are the following: • For every ≤ q ≤ k , the vertex w q is incident to w q − and w q +1 (if they exist). • The vertex w k = v i has neighbors in RS ( a j ) where a j is the origin root section of v i (in particular j < i in the order inherited from b ). • The vertex w is incident to c a i b .Proof. Claim 7 ensures that every vertex is either in RS ( A ) or in F S ( A, B ). By Lemma 25, there is no edgebetween two free sections. So an edge leaving
F S ( a i , b ) has an endpoint in RS ( A ). By definition of incomingvertex, no vertex of F S ( a i , b ) distinct from v i is incident to a vertex of RS ( a j ) with j (cid:54) = i . Moreover, since17 a i b is a minimum path, w is the unique vertex of F S ( a i , b ) which can be incident to RS ( a i ).By definition of v i , there exist edges between v i and the origin root section of v i , namely RS ( a j ). Sinceevery vertex has at most one origin root section, the second point holds.The vertex w is incident to c a i b since they are consecutive in P a i b . Others neighbors of w in RS ( a i ) must becritical vertices since d ( a i , w ) = (cid:96) − d ( c a i b , a i ) = (cid:96) − P a i b is minimum). Thus the matchingproperty of Observation 22 ensures that w has no other neighbor in RS ( a i ), which concludes the proof ofClaim 8.In particular, Claim 8 ensures that any path P leaving F S ( a i , b ) has to enter in RS ( a i ) or in RS ( a j ).Conversely, you can notice that any neighbor of a vertex in RS ( a i ) is either in RS ( a i ) or is in some F S ( a j , b (cid:48) )for j > i . These two observations are the most important pieces of the proofs of the remaining statements.Remind that any path of length at most 2 (cid:96) − a to b does not pass through c a (cid:48) b (cid:48) with b (cid:54) = b (cid:48) . Indeedby independence, d ( a, c a (cid:48) b (cid:48) ) ≥ (cid:96) − d ( b, c a (cid:48) b (cid:48) ) > (cid:96) . Claim 9.
Any path of length at most (cid:96) − from a i to b does not contain any vertex in RS ( a j ) for j > i (in the order inherited from b ).Proof. Assume by contradiction that such a path P exists and denote by j the maximum index such that P passes through RS ( a j ). Note that j ≥
2. Let u be the first vertex of P in RS ( a j ) and let v be the vertexbefore u in P . The path P cannot enter in RS ( a j ) through c a j b (cid:48) with b (cid:48) (cid:54) = b since P has length at most2 (cid:96) −
3. Lemma 21(a) ensures that v / ∈ RS ( A ). So Claim 7 ensures that v ∈ F S ( a k , b (cid:48) ).Assume first that a k (cid:54) = a j . Since u ∈ RS ( a j ), uv is an escape from a j to a k for b (cid:48) . In particular, itmeans that k > j . Since u is the first vertex of P in RS ( a j ), the path P cannot enter in F S ( a k , b (cid:48) ) through RS ( a j ). So Claim 8 ensures that P enters in F S ( a k , b (cid:48) ) through RS ( a k ), contradicting the maximality of j .Assume now that a k = a j . Claim 8 ensures that the unique vertex of F S ( a j , b (cid:48) ) with a neighbor in RS ( a j ) is the first vertex of F S ( a j , b (cid:48) ), so v is this vertex. Moreover, the unique neighbor of v in RS ( a j )is the vertex c a j b (cid:48) by Claim 8. Since P cannot pass through c a (cid:48) b (cid:48) with a (cid:48) (cid:54) = a i and b (cid:48) (cid:54) = b , we have b (cid:48) = b .So u = c a j b and v is the first vertex of F S ( a j , b ). Let us now denote by w the last vertex of P in RS ( a j ).Note that w (cid:54) = c a j b (cid:48) for b (cid:48) (cid:54) = b . Moreover the vertex after w in P cannot be in F S ( a j , b ) since otherwise thisvertex would be v , and then P would not be a path ( v would appear twice in P ). So the edge used to live RS ( a j ) is an escape to a (cid:96) for b (cid:48)(cid:48) . In particular, (cid:96) > j . By Claim 8, vertices of F S ( a (cid:96) , b (cid:48)(cid:48) ) only have neighborsin RS ( a j ) and in RS ( a (cid:96) ). Since w is the last vertex in RS ( a j ), when P leaves F S ( a (cid:96) , b (cid:48) ) it enters in RS ( a (cid:96) ),contradicting the maximality of j . Claim 10.
The vertex c ab is in every path P from a to b of length at most (cid:96) − . Moreover if a vertex ofthe ac ab -subpath of P is not in RS ( a ) , then the next one is.Proof. Let P be a path from a to b of length at most 2 (cid:96) −
3. Let u be the last vertex of u in RS ( a ). Let v bethe vertex after u in P . For distance reasons, P the vertex u is not c ab (cid:48) for b (cid:48) (cid:54) = b . Let us show that P doesnot leave RS ( a ) using an escape. Assume by contradiction that v is in F S ( a j , b (cid:48) ) for a j > a (in the orderinherited from b (cid:48) ). Let us denote by w the first vertex of P after v which is not in F S ( a j , b (cid:48) ). By Claim 8, w is either in RS ( a ) or is c a j b (cid:48) . Since w is after u in P , w / ∈ RS ( a ), so w = c a j b (cid:48) . If b (cid:54) = b (cid:48) , we have a distancecontradiction since both a and b are at distance more than (cid:96) from c a j b (cid:48) . If b = b (cid:48) , then a j > a in the orderinherited from b , contradicting Claim 9.So the vertex u is the vertex c ab . In addition, in the ac ab -subpath of P , if a vertex is not in RS ( a ), thenit is in F S ( a j , b ) where a j > a . Claims 8 and 9 ensure that the next vertex is in RS ( a ).The jump private part of a and b , denoted by JP P ( a, b ), is the set of vertices which are in the jump pathof ab and in no other jump path. Claim 11.
All the paths of length at most (cid:96) − from a to b pass through JP P ( a, b ) . a bb u vc ab c a b Figure 9: Illustration of Claim 11. The two dotted paths are the two sides of an inequality. And the two dashed paths are thetwo sides of the other one.
Proof.
Let P be a path from a to b of length at most 2 (cid:96) −
3. Claim 10 ensures that P passes through c ab .Assume by contradiction that the subpath of P between a and c ab does not pass through JP P ( a, b ). Let u be the last vertex of the ac ab -subpath of P which is in the path P ab (cid:48) for b (cid:48) (cid:54) = b . Such a vertex exists since c ab is not in P ab (cid:48) for b (cid:48) (cid:54) = b by Observation 22. Let v be the vertex after u on P . And, for every b (cid:48) ∈ B , a isin the path P ab (cid:48) .If v / ∈ RS ( a ) then Claim 10 ensures that the vertex after v is in RS ( a ). So v is in P a (cid:48) b (cid:48)(cid:48) with a (cid:48) (cid:54) = a . Thevertex after v is in P ab (cid:48)(cid:48) since it is in RS ( a ). By maximality of u , we have b = b (cid:48)(cid:48) . Thus u ∈ P ab (cid:48) for b (cid:48) (cid:54) = b (by definition of u ) and v ∈ P a (cid:48) b for a (cid:48) (cid:54) = a , a contradiction with Lemma 21(a).So v ∈ RS ( a ) and then v ∈ P ab . Assume by contradiction that v / ∈ JP P ( a, b ). So the vertex v is in thejump path of a (cid:48) b for some a (cid:48) (cid:54) = a . Free to modify a (cid:48) , we may assume that the jump path of a (cid:48) b has beenrerouted only once before v . The vertex v is on the c a (cid:48) b b -subpath of the jump path of a (cid:48) b and u is on the ac ab (cid:48) -subpath of P ab (cid:48) . The two following inequalities, illustrated on Figure 9, provide a contradiction.First d ( u, c ab (cid:48) ) + 3 < d ( v, c a (cid:48) b ) + 1 since d ( a, c ab (cid:48) ) ≤ (cid:96) − d ( a, c a (cid:48) b ) > (cid:96) . Indeed, by definition ofcritical vertex, d ( a, c ab (cid:48) = (cid:96) − d ( a, c a (cid:48) b ) > (cid:96) is a consequence of theindependence. Since u is on a minimum ac ab (cid:48) -path, the inequality holds.Second d ( v, c a (cid:48) b ) < d ( u, c ab (cid:48) ) + 1 since d ( b, c a (cid:48) b ) ≤ (cid:96) and d ( b, c ab (cid:48) ) > (cid:96) and uv is an edge. The first inequalityis due to the fact that jump paths have length at most 2 (cid:96) − a (cid:48) c a (cid:48) b -subpath ofthe jump path of a (cid:48) b is exactly (cid:96) −
3. The second inequality is a consequence of the independence of
A, B .The sum of these two inequalities gives 3 <
2, a contradiction.To conclude the proof of Lemma 26, we apply Theorem 15 with the sets
JP P ( a, b ) for paths of lengthat most 2 (cid:96) −
3. Equation (1) ensures that the jump path of xz has length at most 2 (cid:96) −
3, so if
JP P ( a, b ) isnot selected, there remain paths of length at most 2 (cid:96) −
3. The sets
JP P ( x, z ) are pairwise disjoint and areonly on the jump path of xz . Claims 10 and 11 ensure that the sets JP P ( x, z ) are (2 (cid:96) − X, Z . So the graph G has distance VC-dimension at least d + 1, a contradiction.By combining Theorem 12 and Lemmas 18, 20, 24, 26, we obtain Theorem 11.
5. Concluding remarks
In Section 4, we did not make any attempt to improve the gap function. We made exponential extractionsat several steps as Ramsey’s extractions and the function of Theorem 12 is not expressed in the originalpaper of Matouˇsek. Finding a polynomial gap instead of an exponential one is an interesting problem, thoughprobably a hard one. We can also study this problem for particular classes of graphs. Chepoi, Estellon andVax`es conjectured that the gap function between ν (cid:96) and τ (cid:96) for planar graphs is linear. More formally theyconjectured the following. Conjecture 27. (Chepoi, Estellon, Vax`es [5]) There exists a constant c such that τ (cid:96) ( G ) ≤ c · ν (cid:96) ( G ) for every (cid:96) and every planar graph G . τ (cid:96) ≤ c ( (cid:96) ) ν (cid:96) for bounded expansion classes. Moreover the function c is apolynomial function.In graph coloring, we need some structure to bound the chromatic number. The chromatic number χ ( G )of the graph G is the minimum number of colors needed to color properly the vertices of G , i.e such thattwo adjacent vertices of G receive distinct colors. The size of the maximum clique of G , denoted by ω ( G ),is a lower bound on the chromatic number χ ( G ). The gap between χ and ω can be arbitrarily large sincethere exist triangle-free graphs with an arbitrarily large chromatic number (Erd˝os was the first to constructsome of them in [10]). A class of graphs C is χ -bounded if there exists a function f such that for every graph G ∈ C , every induced subgraph G (cid:48) of G satisfies χ ( G (cid:48) ) ≤ f ( ω ( G (cid:48) )). Dvˇor´ak and Kr´a ’l proved in [7] thatgraphs of bounded rankwidth are χ -bounded. Actually they proved it for classes of graphs with cuts of smallrank. Since the distance VC-dimension catches the complexity of the intersection of neighborhoods at largedistance, the same might be extended for graphs of bounded distance VC-dimension. Conjecture 28.
Let G be a class of graphs. If there exists a function f such that the distance VC-dimensionof G ∈ G is at most f ( ω ( G )) then G is χ -bounded. We also conjecture that the following graph classes, known to be χ -bounded, have a bounded distanceVC-dimension. Conjecture 29.
The distance VC-dimension of every P (cid:96) -free graph G is bounded by a function of (cid:96) and ω ( G ) . Similarly the distance VC-dimension of every circle graph G is bounded by a function of ω ( G ) . Acknowledgement:
The authors want to thank the anonymous reviewers for fruitful comments. We alsowant to thanks L´aszl´o Kozma and Shay Moran for helpful remarks about Theorem 12.
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