Lossy Kernels for Connected Dominating Set on Sparse Graphs

Given a graph $G$ and $k\in{\mathbb N}$, the Dominating Set problem asks for a subset $D$ of $k$ vertices such that every vertex in $G$ is either in $D$ or has a neighbor in $D$. It is well known that Dominating Set is ${\sf W}[2]$-hard when parameterized by $k$. But it admits a linear kernel on graphs of bounded expansion and a polynomial kernel on $K_{d,d}$-free graphs, for a fixed constant $d$. In contrast, the closely related Connected Dominating Set problem (where $G[D]$ is required to be connected) is known not to admit such kernels unless $\textsf{NP} \subseteq \textsf{coNP/poly}$. We show that even though the kernelization complexity of Dominating Set and Connected Dominating Set diverges on sparse graphs this divergence is not as extreme as kernelization lower bounds suggest. To do so, we study the Connected Dominating Set problem under the recently introduced framework of lossy kernelization. In this framework, for $\alpha>1$, an $\alpha$-approximate bikernel (kernel) is a polynomial-time algorithm that takes as input an instance $(I,k)$ and outputs an instance $(I',k')$ of a problem (the same problem) such that, for every $c>1$, a $c$-approximate solution for the new instance can be turned into a $c\alpha$-approximate solution of the original instance in polynomial time. Moreover, the size of $(I',k')$ is bounded by a function of $k$ and $\alpha$. We show that Connected Dominating Set admits an $\alpha$-approximate bikernel on graphs of bounded expansion and an $\alpha$-approximate kernel on $K_{d,d}$-free graphs, for every $\alpha>1$. For $K_{d,d}$-free graphs we obtain instances of size $k^{\mathcal{O}(d^2 / (\alpha-1))}$ while for bounded expansion graphs we obtain instances of size $\mathcal{O}(f(\alpha) k)$ (i.e, linear in $k$), where $f(\alpha)$ is a computable function depending only on $\alpha$.


Introduction
Parameterized complexity and (lossy) kernelization As many real-world computational problems are NP-hard, the first question an algorithm designer should answer when faced with large instances of such problems is whether polynomial-time preprocessing can be applied. Ideally, these preprocessing routines should run very fast (at least faster than the algorithms for solving the problem) and little to no loss of information about solutions should occur. In the past decade or so, progress in parameterized complexity [12] provided a XX:3 which admits an O(log n)-approximation algorithm [22], linear kernels are only known for planar [27] and H-topological-minor-free graphs [18]. Polynomial kernels are excluded already for graphs of bounded degeneracy [8] and for graphs of bounded expansion [13]. In this paper we prove the following theorem. Unfortunately, there is a minor technical detail (which also can be found in [13] for the case of r-Dominating Set) that prevents us from obtaining a reduced instance of the same problem when dealing with graphs of bounded expansion. Instead, we kernelize to an annotated version of the problem, where only a given subset of vertices of G needs to be dominated by a connected set. Although the technical details for dealing with the two graph classes are quite different, the high-level approach is identical. Both kernelization algorithms follow the same two-step strategy. First, our goal is to compute a "small" set of vertices whose domination is sufficient, i.e. the set of dominatees or the so-called domination core. The exact definition of a domination core will be different for the two graph classes but intuitively it is simply a set of vertices whose domination guarantees the domination of the whole graph. Having found a domination core of appropriate size, the next step is to reduce the number of dominators, i.e. vertices whose role is to dominate other vertices, and the number of connectors, i.e. vertices whose role is to connect the solution. When reducing the number of vertices outside the domination core, we borrow approximation techniques that are closely related to the Steiner Tree problem. In the Steiner Tree problem, we are given as input a graph G and a set T ⊆ V (G) of terminals. The objective is to find a subtree of G spanning T that minimizes the number of vertices. Intuitively, we find it easy to think about our approach as follows. The first step borrows ideas from kernelization algorithms for Dominating Set and the second step borrows ideas from approximation algorithms for the Steiner Tree problem. By carefully combining the two and proving additional combinatorial results, we obtain the claimed lossy kernels. Note that the size of our kernel for graphs of bounded expansion matches the size of the best known kernel for Dominating Set, while for K d,d -free graphs we have an additional 1 multiplicative factor in the exponent. This leads us to the following two interesting open questions. Is it possible to reduce the size of our kernel on K d,d -free graphs to f ( )k O(d 2 ) for some function f ? And, in light of the O(k (d−1)(d−3)− ) lower bound for Dominating Set, it is possible to obtain a lossy kernel for Dominating Set on biclique-free graphs that beats this bound? Our hope is that such a "fine-grained" analysis of the kernelization complexity of domination problems will lead to deeper insights and a better understanding of the boundary between "hard" and "easy" instances.

Basics of graphs, sparse graphs, and notations
We use standard terminology from the book of Diestel [11] for those graph-related terms that are not explicitly defined here. All graphs we consider are finite, simple, and undirected. For a graph G, V (G) and E(G) denote the vertex and edge sets of the graph, respectively. We let |G| = |V (G)| denote the number of vertices in G and G = |E(G)| denote the number of edges in G. The density of a graph G, density(G), is defined as density(G) = G /|G|.
The radius of G, denoted radius(G), is the minimum integer r such that there exists a vertex v ∈ V (G), called the center, which is at distance at most r from all vertices in V (G), i.e. the shortest path between v and any other vertex of G has at most r − 1 internal vertices.
When the graph is clear from context we omit the subscript G. We sometimes use the notation For a positive integer r and a vertex v ∈ V (G), we let N r G (v) denote a ball of radius r around v, i.e. the set of all vertices in G that are at distance at most r from v. For a vertex subset X ⊆ V (G), G[X] and G − X are the graphs induced on X and V (G) \ X, Given a graph G and two vertex subsets D, Z ⊆ V (G), we say that D is a Z-dominator if D dominates Z in G, i.e. every vertex z ∈ Z \ D is at distance at most one from some vertex in D. A vertex v dominates itself and all its neighbors. We denote by ds(G, Z) (cds(G, Z)) the size of a smallest (connected) Z-dominator in G. By ds(G) (cds(G)) we mean ds(G, V (G)) (cds(G, V (G))). For a set T ⊆ V (G), we denote by st G (T ) the size of, i.e. the number of vertices in, the smallest Steiner tree connecting T in G (including vertices in T ).
has at most p connected components. Then a set Q ⊆ Z of size at most 2p such that G[D ∪ Q] is connected, can be computed in polynomial time.
is -independent in G if for every pair of distinct vertices in S the distance between them is at least + 1 in G. A useful observation which follows from this definition is that if there exists a 2-independent set S of size k in G, then every dominating set of G must have size at least k since every vertex of G can dominate at most one vertex of S. A clique in a graph is a subset of pairwise adjacent vertices. An independent set is a subset of pairwise non-adjacent vertices. We let K c and I c denote a clique on c vertices and an independent set on c vertices, respectively. We let K i,j denote a biclique (a complete bipartite graph) with i vertices in one partition and j vertices in the other.

Sparse graphs
We define the main classes we consider. We note here that many well-known sparse graph classes such as planar graphs, graph classes with bounded genus, treewidth, or degree as well as graph classes characterizable by finite set of forbidden minors have all bounded expansion and bounded degeneracy. We refer the reader to [6,32] for more details.

Definition 2.2 (Shallow minors).
A graph M is an r-shallow minor of G, for some integer r, if there exists a family of disjoint subsets V 1 , . . . , V |M | of V (G) such that: each graph G[V i ] is connected and has radius at most r, and there is a bijection ω : V (M ) → {V 1 , . . . , V |M | } such that for every edge uv ∈ E(M ) there is an edge in G with one endpoint in ω(u) and another in ω(v).
The set of all r-shallow minors of a graph G is denoted by G r. The set of all r-shallow minors of all members of a graph class G is denoted by G r = G∈G (G r).

Definition 2.3 (Grad and bounded expansion [31]
). For a graph G and an integer r ≥ 0, the greatest reduced average density (grad) at depth r is, For ease of presentation, and since we deal with several constants, we shall use the following convention. We assume that a graph class of bounded expansion G is fixed, and hence so are the values of ∇ i (G), for all non-negative integers i. This assumption is not strictly required but it significantly simplifies the analysis.
Recall that a class of graphs G has bounded degeneracy if every induced subgraph of a graph G ∈ G has a vertex of degree at most d, for some constant d. Note that bounded expansion implies bounded degeneracy, since the degeneracy of a graph G is at most 2∇ 0 (G) (as G 0 contains exactly the subgraphs of G). Moreover a d-degenerate graph cannot contain K d+1,d+1 as a subgraph, which brings us to the class of biclique-free graphs.

Definition 2.4. A class of graphs
Again, we assume a fixed biclique-free graph class, hence a fixed value of d. All of our arguments can be easily extended to K i,j -free graphs, for non-negative integers i and j, but we use K d,d -freeness for simplicity. Next, we state some useful properties of graphs of bounded expansion and biclique-free graphs that will be used later on.
Let G be a graph and X be a subset of its vertices. For u ∈ V (G) \ X, we define the r-projection of u onto X as follows: M G r (u, X) is the set of all vertices w ∈ X for which there exists a path in G that starts in u, ends in w, has length at most r, and whose internal vertices do not belong to X. Note that M G 1 (u, X) = N X (u). We omit the superscript when the graph is clear from context. Lemma 2.7 ([13]). Let G be a class of graphs of bounded expansion. There exists a polynomial-time algorithm that, given a graph G ∈ G, X ⊆ V (G), and an integer r ≥ 1, computes the r-closure of X, denoted by cl r (X), with the following properties: (i) X ⊆ cl r (X) ⊆ V (G), (ii) |cl r (X)| ≤ C cl1 · |X|, and (iii) |M G r (u, cl r (X))| ≤ C cl2 for each u ∈ V (G) \ cl r (X), where C cl1 and C cl2 are constants depending only on r and a fixed (finite) number of grads of G. Lemma 2.8 ([13, 20, 26]). Let G be a class of graphs of bounded expansion and r ∈ N. Let G ∈ G be a graph and let X ⊆ V (G). Then |{Y | Y = M r (u, X) for u ∈ V (G) \ X}| ≤ C ex · |X|, where C ex is a constant depending only on r and a fixed (finite) number of grads of G. Lemma 2.9 ([13]). Let G be a bipartite graph with bipartition (X, Y ) that belongs to some graph class G such that ∇ 1 (G) ≥ 1. Moreover, suppose that for every u ∈ Y we have that N (u) = ∅, and that for every distinct u 1 , u 2 ∈ Y we have N (u 1 ) = N (u 2 ), i.e. Y is twin-free. Then there exists a mapping φ : Y → X with the following properties:

Lossy kernelization
Formally, we first require the notion of a parameterized optimization problem, which is the parameterized analogue of an optimization problem in the theory of approximation algorithms. Throughout this paper we talk about only minimization problems and we refer to [29] to see the symmetric definitions related to maximization problems.
The instances of Π are the pairs (I, k) ∈ Σ * × N and a solution to (I, k) is simply a string S ∈ Σ * such that |S| ≤ |I| + k. The value of a solution S is Π(I, k, S). The optimum value of (I, k) is OPT Π (I, k) = min S∈Σ * ,|S|≤|I|+k Π(I, k, S) and we remove the subscript Π if it is clear from the context. An optimum solution for (I, k) is a solution S such that Π(I, k, S) = OPT Π (I, k). Next we come to the notion of an α-approximate polynomial-time preprocessing algorithm for a parameterized minimization problem Π. Definition 2.12. For α > 1, an α-approximate polynomial-time preprocessing algorithm A for a parameterized minimization problem Π, is defined as a pair of polynomial-time algorithms, called the reduction algorithm R A and the solution lifting algorithm, that satisfy the following properties: 1. Given an instance (I, k) of Π, R A computes an instance (I , k ) = R A ((I, k)) of Π. 2. Given (I, k), (I , k ), and a solution S to (I , k ), the solution lifting algorithm produces a solution S to (I, k) such that Π(I,k,S) An α-approximate kernel for Π is an α-approximate polynomial-time preprocessing algorithm A such that the size of A is upper bounded by a function in the input parameter k. In the case of an α-approximate bikernel, the reduction algorithm is allowed to output an instance of any problem. However, the size of the instance still needs to be bounded by a function of the input parameter k. A polynomial-size approximate kernelization scheme (PSAKS) is a family of α-approximate polynomial kernelization algorithms for each α > 1, with size of each bounded by a polynomial in the input parameter. The size of an output instance of a PSAKS, when run on (I, k) with approximation parameter α, must be upper bounded by f (α)k g(α) , for some functions f and g independent of |I| and k. We refer the reader to the work of Lokshtanov et al. [29] for more details and several examples. In this work, we exhibit lossy kernels for Connected Dominating Set (CDS), which is defined as follows, where G denotes the input graph, k ∈ N and D is a subset of its vertices.
Notice that in standard kernelization or fixed parameter tractable algorithms, when we parameterize by solution size k, for a decision version of a minimization problem, we actually do not care about solutions of size more than k. However, we always aim for efficient algorithms, where efficiency is measured in terms of k. Going by the same logic, we set CDS(G, k, D) = k + 1, when |D| ≥ k + 1, so that all connected dominating sets D of cardinality more than k are "equally bad" or indistinguishable. The symbol ∞ is used to distinguish between actual solutions and other strings which are not solutions. For detailed discussion about capping the objective function at k + 1, we refer to [29, page 16]).

Biclique-free graphs
In this section we show that Connected Dominating Set, parameterized by solution size, admits a PSAKS on K d,d -free graphs. More precisely, we show that Connected Dominating Set admits a (1 + )-approximate kernel on at most k O( d 2 ) vertices.
Finding the domination core. We begin by formally introducing the notion of an kdomination core. We restate that all of our results can be easily extended to K i,j -free graphs, for constant positive integers i and j.
Definition 3.1 (k-domination core). Let G be a graph and Z ⊆ V (G). We say that Z is an k-domination core if every set D of size at most k that dominates Z also dominates V (G).
Note that V (G) is a k-domination core. However, our objective is to obtain a k-domination core Z whose size is polynomially bounded in k. To that end, we start with Z = V (G) and repeatedly reduce the size of Z, maintaining a k-domination core throughout.

Lemma 3.2 ( ).
There exists a polynomial-time algorithm that, given a K d,d -free graph G and a k-domination core Z ⊆ V (G) with |Z| > (2d + 1)k d+1 , either correctly concludes that ds(G) > k (and hence cds(G) > k) or finds a vertex z ∈ Z such that Z \ {z} is a k-domination core.

Reducing connectors and dominators.
Armed with a k-domination core Z of size at most (2d + 1)k d+1 , we partition the graph into two sets Z and R = V (G) \ Z. Next, we define the equivalence relation on R as follows: That is, we partition vertices in R according to their neighborhoods in Z. Let R be the set of equivalence classes defined by . Our goal is to bound the size of R. To that end, we construct a bipartite graph H with bipartition (A, B) from G as follows. We add a vertex a z ∈ A for each vertex z ∈ Z and we add a vertex b κ ∈ B for each equivalence class κ of relation . We add an edge a z b κ ∈ E(H) whenever z is in N Z (w), for some w ∈ κ. Finally, delete any isolated vertices in H. It is not hard to see that H is a subgraph of G and hence is K d,d -free. As a direct implication of Lemma 2.10, we can bound the size of B by O(k O(d 2 ) ).

XX:8 Lossy Kernels for Connected Dominating Set
As Z is a k-domination core, to find a dominating set of size at most k it is enough to find a set which dominates Z. Hence for the purpose of domination, it is redundant to pick more than one vertex from an equivalence class in R. However, to get a connected dominating set, we may need to choose more vertices from an equivalence class. The following lemma finds a small set of relevant vertices which "approximately" preserves the connectivity requirements. Proof. Let (G, k) be the input instance, where G is a connected K d,d -free graph and k is a positive integer. We first describe the reduction algorithm R A . Using the algorithm of Lemma 3.4, with inputs (G, k) and , in polynomial time, algorithm R A either concludes that cds(G) > k and outputs (({v}, ∅), 0); or obtains a set Y ⊆ V (G) and outputs (G[Y ], k) as the reduced instance. Let (G , k ) be the reduced instance. When G = ({v}, ∅), the size of the reduced instance is a constant. Otherwise, by Lemma 3.4, we know that |Y | = k O( d 2 ) , which bounds the kernel size.
The solution lifting algorithm works as follows. Given a solution D to the instance (G , k ), if D is not a connected dominating set of G , then the solution lifting algorithm will output ∅. If D is a connected dominating set, then the algorithm returns D if |D | ≤ k and V (G) otherwise. Let D be the output of the solution lifting algorithm.
We prove that the above reduction algorithm together with the solution lifting algorithm constitute a (1 + )-approximate kernel. Note that if D is not a valid solution of G , then ∅ is not a valid solution for G and CDS(G , k , D ) = CDS(G, k, D) = ∞. Hence we can restrict ourselves to the case when D is a connected dominating set of G . First, consider the case where the algorithm of Lemma 3.4 outputs Y ⊆ V (G) and the reduced instance is hence , 0), we can easily verify that the above mentioned approximation guarantee holds.

Graphs of bounded expansion
In this section we show that Connected Dominating Set, parameterized by solution size, admits a (1 + )-approximate bikernel on at most O(f ( ) · k) vertices. The reduced instance will be an instance of Subset Connected Dominating Set (SCDS), defined as follows: The first phase of our algorithm, i.e. finding a domination core, closely follows the work of Drange et al. [13] but requires subtle changes. We fix a graph class G that has bounded expansion and let (G, k) be the input instance of CDS, where G ∈ G and G is connected. We assume that ∇ 0 (G) ≥ 1, otherwise G is a forest and the problem can be solved in linear time.

Finding the domination core
We begin by formally introducing the notion of a c-exchange domination core, which is different from the definition used in the previous section and from the one considered in [13]. Here, c is a fixed constant which we set later.  c-exchange domination core). Let G be a graph and Z be a subset of vertices of G. We say that Z is a c-exchange domination core if for every set X that dominates Z one of the following conditions holds: (1.) X dominates G, or (2.) there exist A ⊆ X and B ⊆ V (G) such that |B| < |A| ≤ c and (X \ A) ∪ B is a set that dominates Z. Moreover the number of connected components of (X \ A) ∪ B is at most the number of connected components of X. In particular, if X is a connected set then (X \ A) ∪ B is also connected.

Proposition 3 ( ).
Let G be a graph, c be a constant, and Z be a c-exchange domination core of G, and X ⊆ V (G) be a Z-dominator. Then, there is a set Y such that (i) |Y | ≤ |X|, (ii) Y dominates G, (iii) Y can be computed from X in polynomial time, and (iv) the number of connected components of Y is at most the number of connected components of X.
Clearly, V (G) is a c-exchange domination core, for any c, but we look for a c-exchange domination core that is linear in k. Hence, we start with Z = V (G) and gradually reduce |Z| by removing one vertex at a time, while maintaining the invariant that Z is a c-exchange domination core. To this end, we need to prove Lemma 4.2. Note that we only remove vertices from Z at this stage (no vertex deletions), and hence the graph remains intact.

Lemma 4.2.
There exists a constant C core > 0 depending only on a fixed (finite) number of grads of G and a polynomial-time algorithm that, given a graph G ∈ G and a c-exchange domination core Z ⊆ V (G) with |Z| > C core · k, either correctly concludes that cds(G) > k or finds a vertex z ∈ Z such that Z \ {z} is still a c-exchange domination core.
The rest of the subsection is dedicated to proving Lemma 4.2. The algorithm of Lemma 4.2 consists of building a structural decomposition of the graph G. More precisely, we identify a small set X that dominates G, so that if X was deleted from the graph, Z would contain a large subset S, which is 2-independent in the remaining graph. Given such a structure, we can argue that in any optimal Z-dominator, vertices of X serve as dominators for almost all the vertices of S. This is because any vertex of V (G) \ X can dominate at most one vertex from S. Since S will be large compared to X, some vertices of S will be indistinguishable from the point of view of domination via X, and these will be precisely the vertices that can be removed from the domination core. The following lemma, which was proved by Drange et al. [13] and builds on work by Dvorak [15], gives us such a decomposition.

Lemma 4.3.
There exists a constant C Z > 0 depending only on a fixed (finite) number of grads of G and a polynomial-time algorithm that, given a graph G ∈ G, an integer k, a constant C S > 0, and a set Z ⊆ V (G) with |Z | > C Z · k, either correctly concludes that ds(G) > k or finds a pair (X , S ) with the following properties: where C X and C M are constants depending only on a fixed number of grads of G.
Proof. First we run the algorithm of Lemma 4.3 for G, k, Z = Z, C Z = C Z , and constant C S ≥ C S , which we fix later in the proof. If the algorithm of Lemma 4.3 concludes that ds(G) > k, we correctly conclude that ds(G) > k. Otherwise, let (X , S ) be the output of the algorithm of Lemma 4.3. Since Z is a c-exchange domination core, we find using Proposition 3, a set Y , such that |Y | ≤ |X | and Y dominates G. We let X = cl 3 (Y ∪ X ) and S = S \ X. Since X is a superset of Y , X satisfies property (2). Note that by Lemma 2.7, X can be computed in polynomial time and it satisfies property (3). Moreover, |X|. Therefore, if we set the constant C S = 2C cl1 (C S + 2C cl1 ), we satisfy all properties.
Note that when ds(G) > k we can also conclude that cds(G) > k. Hence, in the rest of the section we assume that we are given G, Z, and the constructed sets X and S. We let R = V (G) \ X. Using this notation, S is 2-independent in the graph G[R]. Define the following equivalence relation on S: The following lemma follows directly from the proof of Lemma 3.8 by Drange et al. [13].

Lemma 4.5 ( ).
There exists a constant C eq > 0 depending only on a fixed (finite) number of grads of G such that equivalence relation has at most C eq · 3 Ceq · |X| classes.
We can now set the constant C S that is required in Corollary 4.4 and the constant c of the exchange domination core. We let C S = (4C eq + 1) · C eq · 3 Ceq and c = (4C eq + 1). Since we have that |S| > C S · |X|, from Lemma 4.5 and the pigeonhole principle we infer that there is a class κ of relation with |κ| > 4C eq + 1. We can find such class κ in polynomial time by computing the classes of directly from the definition and examining their sizes. We are ready to prove the final lemma of this section: any vertex of κ can be removed from the c-exchange domination core Z (recall that S ⊆ Z), which concludes the proof of Lemma 4.2. We first show that D dominates Z . Towards that, it is enough to show that every vertex in N [W ] is dominated by M 3 (z, X) ⊆ D . By property (2) of Corollary 4.4, X is a dominating set in G. Hence, every vertex in N [W ] = N (W ) ∪ W either belongs to X or has a neighbor in X. Let Y ⊆ X be the set of vertices in X dominating N [W ]. Since every vertex in N [W ] is at distance at most two from some vertex in κ (because each vertex in W is adjacent to a vertex in κ), all vertices in Y are at distance at most three from some vertex in κ. This implies that Y ⊆ M 3 (z, X) and therefore D dominates Z .
Note that |M 3 (z, X)| < |W | ≤ c and therefore D , which still dominates Z , violates condition (2) of Definition 4.1. However, the set D can have more connected components than D has. Therefore, we add additional vertices to D to ensure connectivity and still violate condition (2). For every vertex u ∈ M 3 (z, X) there is a path P u of length at most three between u and z. Let P = u∈M3(z,X) P u and consider the set It remains to show that D has at most as many connected components as D. If a component C in D contains a vertex from W , then each connected component C of C \ W contains a vertex v C ∈ N (W ) and as mentioned before, v C is adjacent to a vertex in M 3 (z, X). But in D all vertices of M 3 (z, X) are in one component due to the vertices of P . Hence all such components of D contribute to only one component of D . On the other hand, if C does not contain a vertex in W , then clearly all vertices of C are also in D and hence C is connected in D as well. It follows that D has at most as many connected components as D, D dominates Z , and |M 3 (z, X) ∪ P | ≤ C eq + 3C eq < |W | = c = 4C eq + 1, contradicting the fact that D did not satisfy condition (2) of Definition 4.1.

Reducing connectors and dominators
Armed with a c-exchange domination core Z whose size is linear in k, our next goal is to reduce the number of connectors and dominators (the number of vertices in V (G) \ Z). To that end, we need the following lemma which is a generalized version of Lemma 2.11 in [13].
where C tc is a constant depending only on r, q, and a finite number of grads of G. Now letŻ be a superset of Z, which we will fix later in Lemma 4.10, such that |Ż| = O(k). We compute Z = cl 1 (Ż) using Lemma 2.7; then we have that |Z | = O(|Ż|) = O(k). Partition V (G) \ Z into equivalence classes with respect to the following relation : For Hence, using the same reasoning as in the proof of Lemma 4.5 we obtain the following. Construct a graphG as follows. Start with G and, for each equivalence class κ of relation , add a new vertex u κ which is connected to all vertices in κ. Let U = κ∈ {u κ }. Our next step is to apply Lemma 4.7 to graphG with X = Z ∪ U , r = 2, and q = 2t (we fix the value of t later). Before we do so, we need to show thatG still has bounded grads.

Lemma 4.9 ( ).
There is a function f : N → R such that for all r we have ∇ r (G) ≤ f (r). G be a class of bounded expansion and (G, k) be an instance of Connected Dominating Set, where G ∈ G. Let C S = (4C eq + 1) · C eq · 3 Ceq and c = (4C eq + 1). Then, for any fixed > 0, there is a polynomial-time algorithm that either concludes that cds(G) > k or outputs a set Y ⊆ V (G) of cardinality O(f ( ) · k), for some function f , and a set Z ⊆ Y , such that (i) Z is a c-exchange domination core in G and (ii)

Lemma 4.10. Let
Proof. We start by designing an algorithm A with the desired properties. Starting with Lemma 4.2, Algorithm A either concludes that cds(G) > k or finds (in polynomial time) a c-exchange domination core Z of size at most O(k). Since Z is a Z-dominator itself, using Proposition 3, we find a set O, such that |O| ≤ |Z| and O dominates G. Moreover, by Proposition 1, we find a connected supersetÖ of O, such that |Ö| ≤ 3|O|. Next, we leṫ Z = Z ∪Ö and compute Z = cl 1 (Ż) using Lemma 2.7; then we have that |Z | = O(k). We partition V (G) \ Z into equivalence classes with respect to (i.e. u v ⇔ M 1 (u, Z ) = M 1 (v, Z )). From Lemma 4.8, we know that the equivalence relation has at most O(k) classes. Let X be the set of size O(k) obtained by applying Lemma 4.7 to graphG with X = Z ∪ U , r = 2, and q = 2t (we fix the value of t later and U is the set defined earlier).
Algorithm We now show that OPT SCDS ((G[Y ], Z), k) ≤ (1 + )OPT CDS (G, k). Consider the graph D * induced by an optimal connected dominating set of G.
Moreover, the size of each subtree T is at most 2t. We define groups on V (G) as follows. Each vertex v ∈ Z belongs to its own unique group q v and each vertex in V (G) \ Z belongs to group q κ , i.e. vertices in the same equivalence class in belong to the same group. We now construct a new family F which consists of replacing each T ∈ F(D * , t) by a set T in G . Let D = T ∈F V (T ). By the previous claim and the fact that T ∈F (D * ,t) |V (T )| ≤ (1 + 1 t )|V (D * )| + 1, we know that |D | ≤ (1 + 1 t )|V (D * )| + 1 and D dominates Z (since we never reduce the number of groups in D ). Moreover, D consists of at most |V (D * )| t + 1 components. Note thatÖ is a connected Z-dominator and therefore the setÖ ∪ Z is also connected. Now, since Z ∪Ö ⊆ Z , D dominates Z ∪Ö and applying Proposition 1, we obtain a connected (Z ∪Ö)-dominator, and hence also Z-dominator, D of size at most 2|V (D * )| t Setting t = 6 implies that D is a (1 + )-approximate solution, for OPT CDS (G, k) ≥ 6 (note that 3 ≤ 2 · OPT CDS (G, k)). We can assume OPT CDS (G, k) ≥ 6 without loss of generality, as otherwise a simple brute-force algorithm runs in polynomial time.
Using the same arguments as in the proof of Theorem 3.5 and replacing Lemma 3.4 by Lemma 4.10, we obtain the following:  or (3), the size of that tree is at least equal to t and at most equal to 2t. For step (4), the size of the added tree is strictly less than t. Combining those two facts, we know that |F(D, t)| is at most |V (D)| t + 1, as the size of the current tree T D is reduced by at least t for all but one subtrees. To prove the last inequality, i.e. T ∈F (D,t) |V (T )| ≤ (1 + 1 t )|V (D)| + 1, we let mult(v) denote the multiplicity of v minus one, i.e. the number of subtrees T ∈ F(D, t) in which v appears minus one. So if a vertex occurs only once, its multiplicity is zero. Observe that T ∈F (D,t) |V (T )| ≤ |V (D)| + v∈V (T D ) mult(v). However, whenever the multiplicity of a vertex increases by one, the size of the running subtree decreases by at least t (step (3)) or the procedure terminates (step (4)). Hence, v∈T D mult(v) ≤ |V (D)| t + 1, and the bound follows.

B
Missing proofs from Section 3 Proof of Lemma 3.2 Proof. We design such an algorithm as follows. If there is no vertex v ∈ V (G) such that the cardinality of the neighborhood of v in Z is at least |Z| k , then the algorithm terminates and declares that ds(G) > k. Otherwise to find z ∈ Z, the algorithm constructs a sequence of sets Z = X 0 ⊇ X 1 ⊇ · · · ⊇ X and a set S = {v 1 , .
We construct the sets Z = X 0 ⊇ X 1 ⊇ · · · ⊇ X and the set S using an iterative procedure. Initially, we set S := ∅ and X 0 : . If no such vertex v i exists, then the algorithm outputs an arbitrary vertex z ∈ X i−1 \ S and stops.
The above procedure will construct a sequence Z = X 0 ⊇ X 1 ⊇ · · · ⊇ X and a set S = {v 1 , . . . , v }. We first claim that < d. Suppose ≥ d. Then there is a complete bipartite subgraph H of G with bipartition {v 1 , . . . , v d } and d as a subgraph, which is a contradiction to the fact that G is K d,d -free. Hence < d. Moreover, since |X | ≥ |Z| (k) ≥ (2d + 1)k 2 , there always exists a vertex z ∈ X \ S that the algorithm can select to output. Now we prove the correctness of the algorithm. Clearly, if there is no vertex v ∈ V (G) such that the cardinality of the neighborhood of v in Z is at least |Z| k , then ds(G) > k and the algorithm declares it correctly. Otherwise let z ∈ X \ S be the output and let Z = Z \ {z}. We need to prove that Z is still a k-domination core. Let D be a set of size at most k that dominates Z . To prove that D is a dominating set in G, it is enough to show that D dominates Z (because Z is a k-domination core). Since D already dominates Z = Z \ {z}, it suffices to show that D dominates z. Notice that every vertex in S is adjacent to z. Hence, to show that D also dominates Z, it is enough to show that D ∩ S = ∅. We will show that if D ∩ S = ∅, then D cannot dominate all of X \ {z} ⊆ Z . Since our algorithm stops at step + 1, we know that every vertex outside S can dominate strictly less than (2d + 1)k d− vertices from X . As |X \ {z}| ≥ (2d + 1)k d+1− , |D| ≤ k and every vertex outside S can dominate at most (2d + 1)k d− − 1 vertices of X , we have that D ∩ S = ∅. This completes the proof of the lemma.

Proof of Lemma 3.4
Proof. Let t ≥ 1 be a constant, which we fix later. We first define the Group Steiner Tree problem. The input consists of an n-vertex graph H and sets Q 1 , Q 2 , . . . , Q t ⊆ V (H) called groups. The task is to find a tree T of minimum size which contains at least one vertex from every group Q i . The Group Steiner Tree problem can be solved in O(2 t · n O(1) )-time using polynomial space [30]. Now we design an algorithm A with the properties claimed in the lemma. Algorithm A will apply Lemma 3.2 repeatedly starting from a k-domination core V (G) and either conclude that cds(G) > k or find a k-domination core Z of size at most (2d + 1)k d+1 (in polynomial time). Now, let Z be a family of groups {{z} | z ∈ Z} and let R be the set of equivalence classes defined by . The set R ∪ Z forms a family of groups of vertices in V (G). For every subset Q = {Q 1 , . . . , Q } ⊆ R ∪ Z of size at most 2t of groups in R ∪ Z, construct a Group Steiner Tree instance on the graph G with groups Q 1 , . . . , Q . Let T Q be the corresponding solution. Note that since t is a constant each instance can be solved in polynomial time. For every instance that we solve, if the size of T Q is at most 2t then we mark the vertices of T Q in G. We denote the set of all marked vertices by T Q . If T Q is not a dominating set in G, then algorithm A declares that cds(G) > k. Otherwise, since G is assumed to be connected, algorithm A runs the polynomial-time algorithm mentioned in is a connected graph. Note that we also marked a solution of Group Steiner Tree for Q = {Q} for every Q ∈ R ∪ Z. Hence T Q contains all of Z and a vertex from each group in R.
We now prove the correctness of the algorithm. Suppose cds(G) ≤ k, then we claim that algorithm A outputs a set Y. If cds(G) ≤ k, then by Lemma 3.2, algorithm A will, as the first step, correctly find a k-domination core Z of size at most (2d + 1)k d+1 and the set of groups R ∪ Z. Let D be the graph induced by a connected dominating set of size at most k in G (i.e. D is a tree). Let D be an arbitrary set of cardinality at most |V (D)| such that for any w ∈ V (D), there is a vertex w ∈ D with the property that {w, w } ⊆ Q ∈ R ∪ Z and w ∈ T Q . That is, if V (D) has a vertex from a group Q in R ∪ Z, then D also has a vertex from group Q which is, in addition, marked by algorithm A. Note that we can construct the set D since T Q contains at least one vertex from each group in R ∪ Z. Proof of the Claim. Notice that Z ∩ V (D) = Z ∩ D and if any vertex in Z is adjacent to a vertex in a group Q, then it is adjacent to all vertices in group Q. This implies that D also dominates Z and since |D | ≤ |V (D)| ≤ k, by the definition of a k-domination core, D is a dominating set in G.
Hence we can conclude that if cds(G) ≤ k, then T Q always dominates G and algorithm A outputs a set Y . Moreover, since Z ⊆ T Q ⊆ Y , we have that Y contains a k-domination core. This concludes the proof of property (i) of the lemma. Now we prove that OPT(G[Y ], k) ≤ (1 + )OPT(G, k). Let D * be the graph induced by a connected dominating set for G of minimum cardinality (i.e. D * is a tree). If |V (D * )| > k, then OPT(G[Y ], k) ≤ (1 + )OPT(G, k) holds trivially. So we assume that |V (D * )| ≤ k. We let F(D * , t) = {T 1 , T 2 , · · · , T m } denote a (D * , t)-covering family. Proposition 2 implies that there exists such a family for which |F(D * , t)| ≤ |V (D * )| t + 1 and T ∈F (D * ,t) |V (T )| ≤ (1 + 1 t )|V (D * )| + 1. Moreover, the size of each connected subgraph T (in this case also subtree) is at most 2t. We construct a new family F from F(D * , t) as follows. For each T ∈ F(D * , t), we replace T by T Q , where Q is the set of groups from R ∪ Z such that Q ∈ Q if and only if V (T ) ∩ Q = ∅ and T Q is the set of marked vertices in an optimal Steiner tree connecting vertices from the groups in Q. Note that the fact that T is of size at most 2t guarantees the existence of T Q (by construction). Moreover, the size of T Q is at most the size of T , since T is also a solution for Group Steiner Tree for Q. Let D F denote the union of all vertices in F . Let D be a subset of D F , of cardinality at most |V (D * )|, such that for any w ∈ V (D * ), there is a vertex w ∈ D with the property that {w, w } ⊆ Q ∈ R ∪ Z and w ∈ D F . That is, if V (D * ) has a vertex from a group Q in R ∪ Z, then D also has a vertex from group Q. Using the same arguments as in the proof of Claim 2.0.1, we know that D is a dominating set in G. This implies that D F ⊇ D is also a dominating set in G. Applying Proposition 1 in G[Y ] (with D F as dominator and since G[Y ] is connected), we obtain a connected dominating set of size at most 2|F(D * , t)|+|D F | ≤ 2|V (D * )|

C
Missing proofs from Section 4

Proof of Proposition 3
Proof. By Definition 4.1, if X dominates G then we are done. Otherwise, there exist A ⊆ X and B ⊆ V (G) such that |B| < |A| ≤ c, X = (X \ A) ∪ B is a set that dominates Z, and the number of connected components of X is at most the number of connected components of X. Moreover, we can find such sets A and B by going through all possibilities in time O(|V (G)| 2c−1 ) (i.e., in polynomial time for fixed c). By applying this argument iteratively on X , we will eventually find a set Y which dominates G. Furthermore, by Definition 4.1, |B| < |A|, and hence the size of the Z-dominator drops by one in each step. Therefore, the required set Y can be found in time O(|V (G)| 2c ).

Proof of Lemma 4.5
Proof. From Lemma 2.8, we know that the number of different 3-projections in X of vertices of R is bounded by C ex · |X|. Observe that for each u ∈ S, we have M 1 (u, X) ⊆ M 2 (u, X) ⊆ M 3 (u, X) and the number of choices for M 3 (u, X) is again at most C ex · |X| (since S ⊆ R). Moreover, since u ∈ R, we have that |M 3 (u, X)| ≤ C M (property (3) of Corollary 4.4). Hence, to define sets M i (u, X) for 1 ≤ i < 3 it suffices, for every w ∈ M 3 (u, X), to choose the smallest index j, 1 ≤ j ≤ 3, such that w ∈ M j (u, X). The number of such choices is at most 3 C M , and hence the claim follows by setting C eq = C ex + C M .

Proof of Lemma 4.7
Proof. First, using Lemma 2.7 we compute X 0 = cl rq (X). Then, |X 0 | ≤ C cl1 · |X| and for each vertex u / ∈ X 0 we have |M G rq (u, X 0 )| ≤ C cl2 . Now, for each set Y ⊆ X 0 of at most q vertices, compute an optimal Steiner tree T Y whose edges do not belong to G[X 0 ]; in case there is no such tree, set T Y = ∅. Note that T Y can be computed in polynomial time for any fixed q [3]. Define X to be X 0 plus the vertex sets of all trees T Y that have size at most rq.
Claim 3.0.1. |X | ≤ C tc · |X 0 |, where C tc is a constant depending only on r, q, and a finite number of grads of G.

of
... G. From the mapping φ, we can associate each (except at most one) equivalence class with some vertex in Z . In addition, every vertex in Z is associated with at most C ch classes. In other words, when each vertex in Z is replaced by C ch copies, each equivalence obtains a distinct universal vertex. The extra universal vertex added to G K C ch guarantees that the equivalence class with no neighbors in Z is also covered. This completes the proof.

Proof of Claim 4.10.1
Proof. Recall that, when constructing the graphG, we added one universal vertex for each equivalence class in V (G) \ Z . Hence, after applying Lemma 4.7, we know that for any Y ⊆ Z ∪ U of size at most 2t (which is exactly a subset of the groups) if st G (Y ) ≤ 4t then stG(Y ) = st G (Y ). Every vertex in Z belongs to a distinct group and every vertex u κ ∈ U is connected to all vertices in κ. Hence, any tree of size greater than one containing a vertex u κ must also contain a neighbor of u κ (from the same group). The existence of T implies that there exists a tree of size at most 2t connecting all groups appearing in T . Hence, a tree T certifying this fact exists in G .