Contraction Bidimensionality of Geometric Intersection Graphs

Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$ can be contracted to the uniformly triangulated grid $\Gamma_{k}$. A graph class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$ has treewidth $\mathcal{O}({\bf bcg}(G)^{c})$ for some $1\leq c<2$. The SQG${\bf C}$ property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a wide family of graph classes that satisfy the SQG${\bf C}$ property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.


Introduction
Treewidth is one of most well-studied parameters in graph algorithms. It serves as a measure of how close a graph is to the topological structure of a tree (see Section 2 for the formal definition). Gavril is the first to introduce the concept in [30] but it obtained its name in the second paper of the Graph Minors series of Robertson and Seymour in [38]. Treewidth has extensively used in graph algorithm design due to the fact that a wide class of intractable problems in graphs becomes tractable when restricted on graphs of bounded treewidth [1,5,6]. Before we present some key combinatorial properties of treewidth, we need some definitions.
Graph contractions and minors. Our first aim is to define two parameterized versions of the contraction relation on graphs.
Definition 1 (Contractions). Given a non-negative integer c, two graphs H and G, and a surjection σ :  Subquadratic grid minor/contraction property. In order to present the meta-algorithmic potential of bidimensionality theory we need to define some property on graph classes that defines the horizon of its applicability.
Definition 4 (SQGC and SQGC). Let G be a graph class. We say that G has the subquadratic grid minor property (SQGM property for short) if there exist a constant 1 ≤ c < 2 such that every graph G ∈ G which excludes t as a minor, for some integer t, has treewidth O(t c ). In other words, this property holds for G if Proposition 2 can be proven for a sub-quadratic f on the graphs of G.
Similarly, we say that G has the subquadratic grid contraction property (SQGC property for short) if there exist a constant 1 ≤ c < 2 such that every graph G ∈ G which excludes Γ t as a contraction, for some integer t, has treewidth O(t c ). For brevity we say that G ∈ SQGM(c) (resp. G ∈ SQGC(c)) if G has the SQGM (resp SQGC) property for c. Notice that SQGC(c) ⊆ SQGM(c) for every 1 ≤ c < 2.

Algorithmic implications
The meta-algorithmic consequences of bidimensionality theory are summarised as follows. Let G ∈ SQGM(c), for 1 ≤ c < 2, and let p be a minor-bidimensional-optimization parameter.
[A] As it was observed in [10], the problem Π p can be solved in 2 o(k) · n O(1) steps on G, given that the computation of p can be done in 2 O(tw(G)) · n O(1) steps (here tw(G) is the treewidth of the input graph G). This last condition can be implied by a purely meta-algorithmic condition that is based on some variant of Modal Logic [37]. There is a wealth of results that yield the last condition for various optimization problems either in classes satisfying the SQGM propety [18,18,19,41,42] or to general graphs [3,7,24].
[B] As it was shown in [26] (see also [27]), when the predicate φ can be expressed in Counting Monadic Second Order Logic (CMSOL) and p satisfies some additional combinatorial property called separability, then the problem Π p admits a linear kernel, that is a polynomial-time algorithm that transforms (G, k) to an equivalent instance (G , k ) of Π p where G has size O(k) and k ≤ k.
[C] It was proved in [22] (see also [25] and [28]), that the problem of computing p(G) for G ∈ G admits a Efficient Polynomial Approximation Scheme (EPTAS) -that is an -approximation algorithm running in f ( 1 ) · n O(1) steps -given that G is hereditary and p satisfies the separability property and some reducibility property (related to CMSOL expresibility).
All above results have their counterparts for contraction-bidimensional problems with the difference that one should instead demand that G ∈ SQGC(c). Clearly, the applicability of all above results is delimited by the SQGM/SQGC property. This is schematically depicted in Figure 2, where the green triangles Figure 2: The applicability of bidimensionality theory. The green lines correspond the consequences [32] while the red lines correspond to the result of this paper.
triangles indicate the applicability of minor-bidimensionality and the red triangle indicate the applicability of contraction-bidimensionality. The aforementioned Ω(k 2 ·log k) lower bound to the function f of Proposition 2, indicates that SQGM(c) does not contain all graphs (given that c < 2).
As an example we mention the well known d-Domination Set problem (for some d ≥ 1), asking whether a graph G has a set S of at most k vertices such that every vertex in G is within distance at most d from some vertex of S. d-Domination Set is contraction bidimensional problem that satisfies the additional metaalgorithmic conditions in [A], [B], and [C]. This implies that it can be solved in 2 O( √ k) · n time, it admits a linear kernel, and its optimization version admits an EPTAS on every graph class that has the SQGC property.
The emerging direction of research is to detect the most general classes in SQGM(c) and SQGC(c). Concerning the SQGM property, the following result was proven in [15].

Proposition 5. For every graph H, excl(H) ∈ SQGM(1).
A graph H is an apex graph if it contains a vertex whose removal from H results to a planar graph. For the SQGC property, the following counterpart of Proposition 5 was proven in [21]. Proposition 6. For every apex graph H, excl(H) ∈ SQGC(1).
Notice that both above results concern graph classes that are defined by excluding some graph as a minor. For such graphs, Proposition 6 is indeed optimal. To see this, consider K h -minor free graphs where h ≥ 6 (these graphs are not apex graphs). Such classes do not satisfy the SQGC property: take Γ k , add a new vertex, and make it adjacent, with all its vertices. The resulting graph excludes Γ k as a contraction and has treewidth > k.

String graphs
An important step extending the applicability of bidimensionality theory further than H-minor free graphs, was done in [23] (see also [25]). Definition 7 (String graphs, map graphs, and unit disk graphs). Unit disk graphs are intersections graphs of unit disks in the plane and map graphs are intersection graphs of face boundaries of planar graph embeddings. We denote by U d the set of unit disk graphs (resp. of M d map graphs) of maximum degree d.

Proposition 8.
For every positive integer d, U d ∈ SQGM(1) and M d ∈ SQGM(1). Proposition 8 was further extended for intersection graphs of more general geometric objects (in 2 dimensions) in [32]. To explain the results of [32] we need to define a more general model of intersection graphs. Definition 9. (String graphs) Let L = {L 1 , . . . , L k } be a collection of lines in the plane. We say that L is normal if there is no point belonging to more than two lines. The intersection graph G L of L, is the graph whose vertex set is L and where, for each i, j where 1 ≤ i < j ≤ k, the edge {L i , L j } has multiplicity |L 1 ∩ L 2 |. We denote by S d the set containing every graph G L where L is a normal collection of lines in the plane and where each vertex of G L has edge-degree at most d. i.e., is incident to at most d edges. We call S d string graphs with edge-degree bounded by d.
It is easy to observe that U d ∪ M d ⊆ S f (d) for some quadratic function f . Indeed, given a graph G in U d , for each unit disk of its representation in the plane, we can create a string that corresponds to the perimeter of the disk. As all the disks are of the same size, the intersection graph of the strings is homeomorphic to G. The same applies for map graphs by considering the boundaries of the faces and creating a string for each of them. Moreover, apart from the classes considered in [23], S d includes a much wider variety of classes of intersection graphs [32]. As an example, consider C d,α as the class of all graphs that are intersection graphs of α-convex bodies 1 in the plane and have edge-degree at most d. In [32], it was proven that C d,α ⊆ S c where c depends (polynomially) on d and α. Another interesting class from [32] is F H,α containing all H-subgraph free intersection graphs of α-fat 2 families of convex bodies. Notice U d can be seen as a special case of both C d,α and F H,α . (See [36] for other examples of classes included in S d .)

Our contribution
Graph class extensions. Definition 10 ((c 1 , c 2 )-extension). Given a class of graph G and two integers c 1 and c 2 , we define the (c 1 , c 2 )-extension of G, denoted by G (c1,c2) , as the set containing every graph H such that there exist a graph G ∈ G and a graph J that satisfy G ≤ (c1) J and H ≤ c2 J (see Figure 3 for a visualization of this construction). Keep in mind that G (c1,c2) and G (c2,c1) are two different graph classes. We also denote by P the class of all planar graphs.
We visualise the idea of the proof of Proposition 12 by some example, depicted in Figure 4. In Lemma 25 we use the same idea for a more general result. Figure 4 motivates the definition of the (c 1 , c 2 )-extension of a graph class. Intuitively, the fact that H ∈ G (c1,c2) expresses the fact that H can be seen as a "bounded" distortion of a graph in G (after a fixed number of "de-contractions" and contractions).  . . , L k } whose intersection graph G L is depicted in the rightmost figure and has maximum edge degree 9 because of line c that meets other lines in 9 points, therefore G L ∈ S 9 . To see why G L ∈ P (1,9) , one may see the leftmost figure are a planar graph P ∈ P where vertices of degree 1 are discarded. The vertices of this planar graph can be seen as the result of the contraction of the red edges (seen as subgraphs of diameter 1) in the graph J in the middle, i.e., P ≤ (1) J. Finally, the intersection graph G L can be seen as a result of the contraction in J of each one of the paths, on at most 9 vertices, to a single vertex. Therefore G L ≤ 9 J, hence G L belongs in the (1, 9)-extension of planar graphs.
Proposition 6, combined with Proposition 11, provided the wider, so far, framework on the applicability of minor-bidimensionality: SQGM(1) contains excl(H) (c1,c2) for every apex graph H and positive integers c 1 , c 2 . As, by Proposition 6, P ∈ SQGC(1), Proposition 11 and Proposition 12 directly classifies in SQGM(1) the graph class S d , and therefore a large family of bounded degree intersection graphs (including U d and M d ). As a result of this, the applicability of bidimensionality theory for minor-bidimensional problems has been extended to much wider families (not necessarily minor-closed) of graph classes of geometric nature [32].

Our main result.
Definition 13 (Intersection graphs). Given a graph G and a set S ⊆ V (G) we say that S is a connected set of G if G[S] is a connected graph. We also define by C(G) the set of all connected subsets of V (G). Given a C ⊆ C(G), we define the intersection graph of C in G, denoted by I G (C), as the graph whose vertex set is C, where two vertices C 1 and C 2 of I G (C) are connected by an edge if C 1 ∩ C 2 = ∅, and, moreover, the multiplicity of the edge {C 1 , C 2 } is equal to |V (C 1 ∩ C 2 )|. Given a graph class G we define the following class of graphs In other words, inter(G) contains all the intersection graphs of the connected vertex subsets of each of the graphs in G. Given a d ∈ N, we define inter d (G) as the set of graphs in inter(G) that have edge-degree at most d.
However, also the degree bound is maintained, as indicated by the following easy lemma. Figure 5: The hierarchy of graph classes where Proposition 11 applies. U d and M d are the bounded-degree unit-disk and map graphs respectively (where the results of [23,25] apply). S d are the bounded-degree string graphs, while inter d (excl(H)) are the bounded-degree intersection graphs of connected sets of H-minor free graphs, where H is an apex graph.
Proof (sketch). We deal with the less trivial statement that inter d (P) ⊆ S O(d 2 ) . For this, let H ∈ inter d (P) such that H = I G (C) for some collection C of connected subsets of V (G), for some G ∈ P. We choose the planar graph G so that |V (G)| + |E(G)| is minimized. This means that for every C ∈ C, G[C] is a tree on at most 2d vertices. If we now replace each tree G[C] by a string "surrounding" it is easy to observe that two such string cannot have more than O(d 2 ) points in common.
Observe that Proposition 11 exhibits some apparent "lack of symmetry" as the assumption is "qualitatively stronger" than the conclusion. This does not permit the application of bidimensionality for contractionbidimensional parameters on classes further than those of apex-minor free graphs. In other words, the results in [32] covered, for the case of S d , the green triangles in Figure 2 but left the red triangles open. The main result of this paper is to fill this gap by proving the following extension of Proposition 11. The main result of this paper is the following.

Consequences.
We call a graph class monotone if it is closed under taking of subgraphs, i.e., every subgraph of a graph in G is also a graph in G. A powerful consequence of Theorem 15 is the following (the proof is postponed in Section 3).
Combining Proposition 6 and Theorem 16 we obtain that SQGC(1) contains inter d (excl(H)) for every apex graph H. This extends the applicability horizon of contraction-bidimensionality further than apex-minor free graphs (see Figure 5). As a (very) special case of this, we have that S d ∈ SQGC(1). Therefore, on S d , the results described in Subsection 1.2 apply for contraction-bidimensional problems as well.
This paper is organized as follows. In Section 2, we give the necessary definitions and some preliminary results. We prove Theorem 16 in Section 3 while Section 4 is dedicated to the proof of Theorem 15. We should stress that this proof is quite different than the one of Proposition 11 in [32]. Finally, Section 5 contains some discussion and open problems.

Definitions and preliminaries
We denote by N the set of all non-negative integers. Given r, q ∈ N, we define [r, q] = {r, . . . , q} and [r] = [1, r].
All graphs in this paper are undirected, loop-less, and may have multiple edges. If a graph has no multiple edges, we call it simple. Given a graph G, we denote by V (G) its vertex set and by E(G) its edge set. Let x be a vertex or an edge of a graph G and likewise for y; their distance in G, denoted by dist G (x, y), is the smallest number of vertices of a path in G that contains them both. Moreover if G is a graph and x ∈ V (G), we denote by N c G (x), for each c ∈ N, the set {y | y ∈ V (G), dist G (x, y) ≤ c + 1}. For any set of vertices S ⊆ V (G), we denote by G[S] the subgraph of G induced by the vertices from S. If G[S] is connected, then we say that S is a connected vertex set of G. We define the diameter of a connected subset S as the maximum pairwise distance between any two vertices of S. The edge-degree of a vertex v ∈ V (G) is the number of edges that are incident to it (multi-edges contribute with their multiplicity to this number).
that have not been added already. For an example of Γ k (resp.Γ k ), see Figure 1 (resp. Figure 6). Notice that Γ k is a triangulation of k . In each of these graphs we denote the vertices of the underlying grid by their coordinates (i, j) ∈ [0, k − 1] 2 agreeing that the upper-left corner (i.e., the unique vertex of degree 3) is the vertex (0, 0).Γ k has two vertices of degree 3, the top left and the bottom right of the grid part. We call Γ k the uniformly triangulated grid andΓ k the extended uniformly triangulated grid.
Definition 18 (Treewidth). A tree-decomposition of a graph G, is a pair (T, X ), where T is a tree and X = {X t : t ∈ V (T )} is a family of subsets of V (G), called bags, such that the following three properties are satisfied: The width of a tree-decomposition is the cardinality of the maximum size bag minus 1 and the treewidth of a graph G is the minimum width over all the tree-decompositions of G. We denote the treewidth of G by tw(G).

Lemma 19. Let G be a graph and let
Proof. By definition, since H is a c-size contraction of G, there is a mapping between each vertex of H and a connected set of at most c edges in G, so that by contracting these edge sets we obtain H from G. The endpoints of these edges form disjoint connected sets in G, implying a partition of the vertices of G into connected sets Consider now a tree decomposition (T, X ) of H. We claim that the pair (T, X ), where X t := x∈Xt V x for t ∈ T is a tree decomposition of G. Clearly all vertices of G are included in some bag, since all vertices of H did. Every edge of G with both endpoints in the same part of the partition is in a bag, as each of these vertex sets is placed as a whole in the same bag. If e is an edge of G with endpoints in different parts of the partition, say V x and V y , then this implies that {x, y} ∈ E(H). Thus, there is a node t of T for which x, y ∈ X t and therefore e ⊆ X t . Moreover, the continuity property remains unaffected, since for any vertex x ∈ V (H) each vertex in V x induces the same subtree in T that x did.
In Table 1 we present all the notation that we use in this paper.

Symbol
Combinatorial object Definition the graph class that is the (c1, c2)-extension of a graph class G 10 Table 1: Graphs, graph classes, and functions of the paper.

Proof of Theorem 16
We start with the following useful property of the contraction relation. We Let v be a vertex in G incident to exactly two edges e 1 = {v, v } and e 2 = {v, v }, and let G be the graph obtained from G after the dissolution of v. Let σ : V (G ) → V (Q) such that for all z ∈ V (G ), σ (z) = σ(z). As the dissolution maintains connectivity, we have that for every is connected. Moreover, as δ(Q) ≥ 3, we know that for each x ∈ V (Q), there exists z ∈ σ −1 (x) such that z has edge degree at least 3. In particular we know that z is different from v. So we have that G[σ −1 (x)] is a non-empty graph. Thus Q ≤ G . The lemma follows by iterating this argument. (The function bcg). Given a graph G, we define bcg(G) as the maximum k for which G can be contracted to the uniformly triangulated grid Γ k .

Definition 21
Notice that bcg is a contraction-closed parameter, i.e., if H ≤ G, then bcg(H) ≤ bcg(G). H and G be two graphs. If H is a dissolution of G, then bcg(H) = bcg(G).

Lemma 22. Let
Proof. The fact that bcg(H) ≤ bcg(G) follows from the fact that H is also a contraction of G and taking into account the contraction-closedness of bcg. The fact that bcg(G) ≤ bcg(H) follows by taking into account that δ(Γ k ) ≥ 3 and applying inductively Lemma 20 to the vertices of degee 2 in G that need to be dissolved in order to transform G to H. Definition 23. Given a graph class G, we define the dissolution closure of G as the graph class diss(G) containing all the dissolutions of the graphs in G.
We observe the following.
Proof. Suppose that G ∈ SQGC(c) for some 1 ≤ c < 2, wich implies that Let H ∈ diss(G) and let G ∈ G such that H is a dissolution of G. By Lemma 22, bcg(H) = bcg(G) and from (1), tw(G) ≤ λ(bcg(H)) c . As H is a minor of G, we have that tw(H) ≤ λ(bcg(H)) c and the lemma follows.
The next lemma uses as a departure point the same idea as the one of proof of Proposition 12, visualized by the example of Figure 4.

Lemma 25. If G is a graph class that is topological minor closed, then inter
Proof. Let H be a graph on h vertices in inter d (G), for some d ∈ N. This means that there is a graph G in G such that we can see the vertices of H as a set C = {C 1 , . . . , C h } of connected subsets of G and each multi-edge e = {C i , C j } of H corresponds to two mutually intersecting subsets of C and the multiplicity of e . Also, a vertex in V cannot belong in more that d + 1 distinct C i 's as, otherwise H would contain a clique with at least d + 2 vertices. As H ∈ inter d (G), this is not possible.
Recall that, for each i ∈ [h], V i is a subset of C i and let T i be a minimum-size tree of G[C i ] containing the vertices of V i . We partition the set of vertices of T i into three sets V i , V i , D i where among the vertices in V (T i ) \ V i , D i are the vertices of degree 2 and V i are the rest. By minimality, the leaves of T i belong in V i . Moreover, there is no vertex in V i that belongs to some other V i , i ∈ [h] \ {i}. We denote byT i the tree obtained from T i if we dissolve in T i all vertices of D i . That way we can still partition the vertices of eachT i , i ∈ [h], into V i and V i . Also, it is easy to see thatT i has diameter at most |V i | − 1 ≤ d − 1.
We define the graph G := i∈[h]T i . Notice that G is obtained from i∈[h] T i (that is a subgraph of G) after we dissolve all vertices in i∈[h] D i . Therefore G is a topological minor of G, thus G ∈ G. We consider the collection T = {T 1 , . . . ,T h } of connected subgraphs of G .
We define the graph J to be the disjoint union of the h trees in T in which, for each x ∈ V , we add a clique between all the copies of x. Notice that each added clique has size at least 2 and at most d + 1.
Observe now that G ≤ (d+1) J, as G is obtained after contracting in J the aforementioned pairwise disjoint cliques. Moreover, H ≤ d−1 J as H is obtained after we contract in J eachT i (of diameter ≤ d − 1) to a single vertex. As G ∈ G, we conclude that H ∈ G (d+1,d−1) as required.
We are now ready to prove Theorem 16.

Proof of Theorem 15
Let H and G be graphs and c be a non-negative integer. If H ≤ c σ G, then we say that H is a σ-contraction of G, and denote this by H ≤ σ G.
Before we proceed the the proof of Theorem 15 we make first the following three observations. (In all statements, we assume that G and H are two graphs and σ : Observation 26. Let S be a connected subset of V (H). Then the set x∈S σ −1 (x) is connected in G.
Observation 28. Let S be a connected subset of V (G). Then the diameter of σ(S) in H is at most the diameter of S in G.
Given a graph G and S 1 , S 2 ⊆ V (G) we say that S 1 and S 2 touch if either S 1 ∩ S 2 = ∅ or there is an edge of G with one endpoint in S 1 and the other in S 2 .
We say that a collection R of paths of a graph is internally disjoint if none of the internal vertices, i.e., none of the vertex of degree 2, of some path in R is a vertex of some other path in R. Let A be a collection of subsets of V (G). We say that A is a connected packing of G if its elements are connected and pairwise disjoint. If additionally A is a partition of V (G), then we say that A is a connected partition of G and if, additionally, all its elements have diameter bounded by some integer c, then we say that A is a c-diameter partition of G.

Λ-state configurations.
Definition 29. (Λ-state configurations) Let G be a graph. Let Λ = (W, E) be a graph whose vertex set is a connected packing of G, i.e., its vertices are connected subsets of V (G). A Λ-state configuration of a graph G is a quadruple S = (X , α, R, β) where 1. X is a connected packing of G, 2. α is a bijection from W to X such that for every W ∈ W, W ⊆ α(W ), 3. R is a collection of internally disjoint paths of G, and 4. β is a bijection from E to R such that if {W 1 , W 2 } ∈ E then the endpoints of β({W 1 , W 2 }) are in W 1 and W 2 and V (β({W 1 , W 2 })) ⊆ α(W 1 ) ∪ α(W 2 ).

Definition 30. (States, freeways, clouds, and coverage) A Λ-state configuration
. We refer to the elements of X as the states of S and to the elements of R as the freeways of S. We define indep(S) = V (G) \ X∈X X. Note that if S is a Λ-state configuration of G, S is complete if and only if indep(S) = ∅. Let A be a c-diameter partition of G. We refer to the sets of A as the A-clouds of G. We define front A (S) as the set of all A-clouds of G that are not subsets of some X ∈ X . Given a A-cloud C and a state X of S we say that C shadows X if C ∩ X = ∅. The coverage cov S (C) of an A-cloud C of G is the number of states of S that are shadowed by C. A Λ-state configuration S = (X , α, R, β) of G is A-normal if its satisfies the following conditions: (B) If a A-cloud over S intersects the vertex set of at least two freeways of S, then it shadows at most one state of S.
We define cost A (S) = C∈front A (S) cov S (C). Given S 1 ⊆ S 2 ⊆ V (G) where S 1 is connected, we define cc G (S 2 , S 1 ) as the (unique) connected component of G[S 2 ] that contains S 1 .

Triangulated grids inside triangulated grids
The next lemma is the main combinatorial engine of our results. We assume that H ≤ c G and Γ k ≤ G.
Here H should be seen as the result of a "shrink" of G in the sense that G can be contracted to H so that each vertex of H is created after a bounded number of contractions. The lemma states that if G can be contracted to a uniformly triangulated grid, then H, as a "shrunk version" of G, can also be contracted to a uniformly triangulated grid that is no less than linearly smaller.
The proof strategy views the graph G as being contracted into a uniformly triangulated grid Γ k (see Figure 7), we choose a scattered set of "capitals" in it (the black vertices in Figure 7). Then we set up a "conquest" procedure where each capital is trying to expand to a country. This procedure has three phases. The first phase is the expansion face where each country tries to incorporate unconquested territories around it (the limits of this expansion is depicted by the red cycles in Figure 7). The second phase is the clash face, where different countries are fighting for disputed territories. Finally, the third phase is the annex phase, where each country naturally incorporates remaining enclaves. The end of this war creates a set of countries occupying the whole G that, when contracted, give rise to a uniformly triangulated grid Γ k , where k = Ω(k).

Lemma 31. Let G and H be graphs and c, k be non-negative integers such that
For each (i, j) ∈ 0, k + 1 2 , we define b i,j to be the vertex of Γ with coordinate (i(2c + 1), j(2c + 1)).
Here b out can be seen as a vertex that "represents" all vertices in Q out .
Let q, p be two different elements of Q. We say that q and p are linked if they both belong in Q in and their distance in Γ is 2c + 1 or one of them is b out and the other is b i,j where i ∈ {1, k } or j ∈ {1, k }.
For each q ∈ Q in , we define W q = σ −1 (q). W q is connected by the definition of σ. In case q = b out we define W q = q ∈Qout σ −1 (q ). Note that as Q out is a connected set of Γ, then, by Observation 26, W bout is connected in G. We also define W = {W q | q ∈ Q}. Given some q ∈ Q we call W q the q-capital of G and a subset S of V (G) is a capital of G if it is the q-capital for some q ∈ Q. Notice that W is a connected packing of V (G).
Let q ∈ Q. If q ∈ Q in then we set N q = N c Γ (q). If q = b out , then we set N q = q ∈Qout N c Γ (q ). Note that for every q ∈ Q, N q ⊆ V (Γ). For every q ∈ Q, we define X q = σ −1 (N q ). Note that X q ⊆ V (G). We also set X = {X q | q ∈ Q}. Let q and p we two linked elements of Q. If both q and p belong to Q in , and therefore are vertices of Γ, then we define Z p,q as the unique shortest path between them in Γ. If p = b out and q ∈ Q in , then we know that q = b i,j where i ∈ {1, k } or j ∈ {1, k }. In this case we define Z p,q as any shortest path in Γ between b i,j and the vertices in Q out . In both cases, we define P p,q by picking some path between W p and W q in G[σ −1 (V (Z p,q ))] such that |V (P p,q ) ∩ W q | = 1 and |V (P p,q ) ∩ W p | = 1.
Let E = {{W p , W q } | p and q are linked} and let Λ = (W, E). Notice that Λ is isomorphic toΓ k and consider the isomorphism that correspond each vertex q = b i,j , i, j ∈ 1, k 2 to the vertex with coordinates (i, j). Moreover b out corresponds to the apex vertex ofΓ k . Let α : W → X such that for every q ∈ Q, α(W q ) = X q . Let also R = {P p,q | p, q ∈ Q, p and q are linked}. We define β : E → R such that if q and p are linked, then β(W q , W p ) = P p,q . We use notation S = (X , α, R, β). Claim 32. S is an A-normal Λ-state configuration of G.
Proof of Claim 32. We first see that S is a Λ-state configuration of G. Condition 1 follows by the definition of X q and Observation 26. Condition 2 follows directly by the definitions of W q and X q . For Condition 3, we first observe that, by the construction of Γ and the definition of Z p,q , for any two pairs p, q and p , q In this whole graph Γ k , we initialize our reaserch of Γ k such that every internal red hexagon will become a vertex ofΓ k and correspond to a state and the border, also circle by a red line will become the vertex b out . The blue edges correspond to the freeways. Red cycles correspond to the boundaries of the starting countries. Blue paths between big-black vertices are the freeways. Big-black vertices are the capitals.
of pairwise linked elements of Q, the paths Z p,q and Z p ,q are internally vertex disjoined paths of Γ. It implies that P p,q and P p ,q can intersect each other only on the vertices of W p ∪ W q ∪ W p ∪ W q . But P p,q (resp. P p ,q ), by construction contains only two vertices of W p ∪ W q ∪ W p ∪ W q that are the extremities of P p,q , (resp. P p ,q ). So P p,q and P p ,q are internally vertex disjoined, as required. For Condition 4, assume that {W p , W q } ∈ E. The fact that the endpoints of β({W p , W q }) are in W p and W q follows directly by the definition of β({W p , W q }) = P p,q . It remains to prove that V (β({W p , W q })) ⊆ α(W p ) ∪ α(W q ) or equivalently, that V (P p,q ) ⊆ X p ∪ X q . Observe that, if both p, q ∈ Q in , then every vertex in the shortest path Z p,q should be within distance c from either p or q. Similarly, if p ∈ Q in and q = b out , then every vertex in the shortest path Z p,q should be within distance c from either p or some vertex in Q out . So for every p, q ∈ Q, with p = q, Z p,q ⊆ N p ∪ N q . By Observation 27, every vertex in σ −1 (V (Z p,q )) belongs to X p ∪ X q and the required follows as V (P p,q ) ⊆ σ −1 (V (Z p,q )). This completes the proof that S is a Λ-state configuration of G.
We now prove that S is A-normal. Recall that A be a c-diameter partition of G. Let C be a A-cloud and let C = σ(C) be a subset of V (Γ). As C is of diameter at most c, then, from Observation 28, C is also of diameter at most c. Notice that if C intersects some member W of W, then C = σ(C) also intersects σ(W ), therefore C intersects some element of Q in ∪ Q out . Assume C contains p ∈ Q in ∪ Q out , then C ⊆ N p . From Observation 26, C ⊆ X p = α(W p ), therefore C satisfies Condition (A).
By construction, the distance in Γ between two elements of Q in is either 2c + 1 or at least 4c + 2. The distance in Γ between on elements of Q in and any element of Q out is a multiple of 2c + 1. This implies that if p, q ∈ Q, p = q, N p ∩ C = ∅, and N q ∩ C = ∅, then p and q are linked.
By construction, if p and q are linked, then for every r ∈ Q and every u ∈ Z p,q , dist Γ (r, u) ≥ min(dist Γ (r, p), dist G (r, q)), where for every x ∈ Q in , the quantity dist Γ (x, b out ) is interpreted as min{dist Γ (x, q ) | q ∈ Q out }. This implies that if C intersects Z p,q for some p, q ∈ Q, then for every r ∈ Q \ {p, q}, then C does not intersect N r . We will use this fact in the next paragraph towards completing the proof of Condition (B).
We now claim that if C intersects two distinct paths in {Z p,q | (p, q) ∈ Q 2 , p = q}, then C intersects at most one of the sets in {N q | q ∈ Q}. Let Z p,q and Z p ,q be two distinct paths intersected by C . We argue first that p, q, p , q cannot be all different. Indeed, if this is the case, as C intersects Z p,q then C cannot intersect N p or N q as p , q ∈ {p, q}. As Z p ,q ⊆ N q ∪ N p , we have a contradiction. Assume now that p = p and q = q . As C intersects Z p,q , then it does not intersect N r for any r ∈ Q \ {p, q}, and as it intersects Z p,q , then it does not intersect N r for any r ∈ Q \ {p, q }. We obtain that C intersects at most one of the sets in {N r | r ∈ Q} that is N p . By definition of the states, we obtain that C shadows at most one state that is X p . That completes the proof of condition (B).
We define bellow three ways to transform a Λ-state configuration of G. In each of them, S = (X , α, R, β) is an A-normal Λ-state configuration of G and C is an A-cloud in front A (S).
1. The expansion procedure applies when C intersects at least two freeways of S. Let X be the state of S shadowed by C (this state is unique because of property (B) of A-normality). We define (X , α , R , β ) = expand(S, C) such that for each W ∈ W, α (W ) = X where X is the unique set of X such that W ⊆ X , -R = R, and β = β.
2. The clash procedure applies when C intersects exactly one freeway P of S. Let X 1 , X 2 be the two states of S that intersect this freeway. Notice that P = β(α −1 (X 1 ), α −1 (X 2 )), as it is the only freeway with vertices in X 1 and X 2 . Assume that (C ∩ V (P )) ∩ X 1 = ∅ (if, not, then swap the roles of X 1 and X 2 ). We define (X , α , R , β ) = clash(S, C) as follows: for every X ∈ X , because of property (A) of A-normality), for each W ∈ W, α (W ) = X where X is the unique set of X such that W ⊆ X , where P = P 1 ∪ P * ∪ P 2 is defined as follows: let s i be the first vertex of C that we meet while traversing P when starting from its endpoint that belongs in W i and let P i the subpath of P that we traversed that way, for i ∈ {1, 2}. We define P * by taking any path between s 1 and s 2 inside G[C], and 3: The annex procedure applies when C intersects no freeway of S and touches some country X ∈ X . We define (X , α , R , β ) = anex(S, C) such that for every X ∈ X , because of property (A) of A-normality), for each W ∈ W, α (W ) = X where X is the unique set of X such that W ⊆ X , -R = R, and β = β.
Proof of Claim 33. We first show that S is an A-normal Λ-state configuration of G. In each case, the construction of S makes sure that X is a connected packing of G and that the countries are updated in a way that their capitals remain inside them. Moreover, the highways are updated so to remain internally disjoint and inside the corresponding updated countries. We next prove that S is A-normal. Condition (A) is invariant as the cloud we take into consideration cannot intersect any W ∈ W and a cloud intersecting some capital W ∈ W cannot be disconnected from W . It now remains to prove condition (B). Because of Condition 4 of the definition of a Λ-state configuration, if a cloud C intersects a freeway, then it shadows at least one state. Now assume that a cloud C intersects two freeways in S , then by construction of S , it also intersects at least the two same freeways in S. This along with the fact that S satisfies Condition (B), implies that S satisfies condition (B) as well, as required. Notice that, for any cloud C * ∈ A \ {C}, if C * does not intersect a state X in S, then the corresponding state X in S , i.e., the state X = α (α −1 (X)), also does not intersect C * . This means that cost(S , A) ≤ cost(S, A).
Notice now that by the construction of S , C is not in front A (S ). In the case where cov S (C) ≥ 1 we have that cost(S , A) < cost(S, A).
Notice that the case where cov S (C) = 0 happens only when action = anex and there is an edge with one endpoint in C and one in some country X * of S that does not intersect C. Moreover cc G (X \C, α −1 (X)) = X, for every state X of S. This implies that indep(S ) ⊆ indep(S). As C ⊆ indep(S) and C ∩ indep(S ) = ∅, we conclude that |indep(S )| < |indep(S)| as required.
To continue with the proof of Lemma 31 we explain how to transform the A-normal Λ-state configuration S of G to a complete one. This is done in two phases. First, as long as there is an A-cloud C ∈ front(S) where cov S (C) ≥ 1, we apply one of the above three procedures depending on the number of freeways intersected by C. We again use S to denote the A-normal Λ-state configuration of G that is created in the end of this first phase. Notice that, as there is no A-cloud with cov S (C) ≥ 1, then cost A (S) = 0. The second phase is the application of anex(S, C), as long as some C ∈ front A (S) is touching some of the countries of S. We claim that this procedure will be applied as long as there are vertices in indep(S). Indeed, if this is the case, the set front A (S) is non-empty and by the connectivity of G, there is always a C ∈ front A (S) that is touching some country of S. Therefore, as cost A (S) = 0 (by Claim 33), procedure anex(S, C) will be applied again.
By Claim 33, |indep(S)| is strictly decreasing during the second phase. We again use S for the final outcome of this second phase. We have that indep(S) = ∅ and we conclude that S is a complete A-normal Λ-state configuration of G such that |front A (S)| = 0.
We are now going to create a graph isomorphic to Λ only by doing contractions in G. For this we use S, a complete A-normal Λ-state configuration of G such that |front A (S)| = 0, obtained as describe before. We contract in G every country of S into a unique vertex. This can be done because the countries of S are connected. Let G be the resulting graph. By construction of S, G is a contraction of H. Because of Condition 4 of Λ-state configuration, every freeway of S becomes an edge in G . This implies that there is a graph isomorphic to Λ that is a subgraph of G . SoΓ k is isomorphic to a subgraph of G with the same number of vertices. Let seeΓ k as a subgraph of G and let e be an edge of G that is not an edge ofΓ k . As e is an edge of G , this implies that in G, there is two states of S such that there is no freeway between them but still an edge. This is not possible by construction of S. We deduce that G is isomorphic toΓ k . Moreover, as |front A (S)| = 0, then every cloud is a subset of a country. This implies that G is also a contraction of H. By contracting in G the edge corresponding to {a, (k − 1, k − 1)} inΓ k , we obtain that Γ k is a contraction of H. Lemma 31 follows.
Let G ∈ SQGC(c) be a class of graph such that ∀G ∈ G tw(G) ≤ λ · (bcg(G)) c . Let H ∈ G (c1,c2) and let G and J be two graphs such that G ∈ G, G ≤ (c1) J, and H ≤ c2 J. G and J exist by definition of G (c1,c2) .
• By definition of H and J, tw(H) ≤ tw(J).
As the formula is independent of the graph class, the Theorem 15 follows.

Conclusions, extensions, and open problems
The main combinatorial result of this paper is that, for every d and every apex-minor-free graph class G, the intersection class inter d (G) has the SQGC property for c = 1. Certainly, the main general question is to detect even wider graph classes with the SQGM/SQGC property. In this direction, some insisting open issues are the following: • Is the bound on the (multi-)degree necessary? Are there classes of intersection graphs with unbounded or "almost bounded" maximum degree that have the SQGM/SQGC property?
• All so far known results classify graph classes in SQGM(1) or SQGC(1). Are there (interesting) graph classes in SQGM(c) or SQGC(c) for some 1 < c < 2 that do not belong in SQGM(1) or SQGC(1) respectively? An easy (but trivial) example of such a class is the class Q d of the q-dimensional grids, i.e., the cartesian products of q ≥ 2 equal length paths. It is easy to see that the maximum k for which an n-vertex graph G ∈ Q q contains a (k × k)-grid as a minor is k = Θ(n 1 2 ). On the other size, it can also be proven that tw(G) = Θ(n q−1 q ). These two facts together imply that Q q ∈ SQGM(2 − 2 q ) while Q q ∈ SQGM(2 − 2 q − ) for every > 0.
• Usually the graph classes in SQGC(1) are characterised by some "flatness" property. For instance, see the results in [31,34,34] for H-minor free graphs, where H is an apex graph. Can SQGC(1) be useful as an intuitive definition of the "flatness" concept? Does this have some geometric interpretation?