Domination Problems in Nowhere-Dense Classes of Graphs

Transcription

1 LIPIcs Leibniz International Proceedings in Informatics Domination Problems in Nowere-Dense Classes of Graps Anuj Dawar 1, Stepan Kreutzer 2 1 University of Cambridge Computer Lab, U.K. anuj.dawar@cl.cam.ac.uk 2 Oxford University Computing Laboratory, U.K. kreutzer@comlab.ox.ac.uk ABSTRACT. We investigate te parameterized complexity of generalisations and variations of te dominating set problem on classes of graps tat are nowere dense. In particular, we sow tat te distance-d dominating-set problem, also known as te (k, d)-centres problem, is fixed-parameter tractable on any class tat is nowere dense and closed under induced subgraps. Tis generalises known results about te dominating set problem on H-minor free classes, classes wit locally excluded minors and classes of graps of bounded expansion. A key feature of our proof is tat it is based simply on te fact tat tese grap classes are uniformly quasi-wide, and does not rely on a structural decomposition. Our result also establises tat te distance-d dominating-set problem is FPT on classes of bounded expansion, answering a question of Nešetřil and Ossona de Mendez. 1 Introduction Te dominating-set problem is among te most well-studied problems in algoritmic grap teory and complexity teory. Given a grap G and an integer k, we are asked to determine weter G contains a set X of at most k vertices suc tat every vertex of G is eiter in X or adjacent to a vertex in X. Tis is a classical NP-complete problem tat as been intensively studied from te point of view of approximation algoritms and fixed-parameter tractability. A number of generalisations and variations of te dominating set problem ave also been studied in tis context. In particular, te distance-d dominating-set problem is one were we are given a grap G and integer parameters d and k and we are to determine weter G contains a set X of at most k vertices suc tat every vertex in G as distance at most d to a vertex in X. Tis problem, also known as te (k, d)-centre problem, as for instance been studied in [5] in connection wit network centres and oter clustering problems (see te references in [10]). It is clear tat in te case d = 1, tis is just te dominating set problem. A number of oter domination problems are considered in Section 5. We are interested in investigating tese problems from te point of view of fixedparameter tractability. Tat is we are interested in algoritms for tese problems tat run in time f (k) n c (or f (k + d) n c in te case of te distance-d dominating-set problem) were n is te order of te grap G, c is independent of te parameters k and d and f is any computable function. Suc algoritms are unlikely to exist in general, since te dominating-set problem is W[2]-complete (see [13, 15] for a general introduction to parameterized complexity, including definitions of FPT and W[2]). However, if we restrict te class of graps under c Dawar, Kreutzer; licensed under Creative Commons License-NC-ND

2 consideration, we can obtain efficient algoritms in te sense of fixed-parameter tractability, even toug te problem remains NP-complete on te restricted class. We are interested in knowing ow general we can make our grap classes wile retaining fixed-parameter tractability. In tis paper, we pus te tractability frontier forward by sowing tat te distance-d dominating-set problem as well as a number of oter domination problems, are FPT on nowere-dense classes of graps. Tis generalises known results about te dominating set problem on H-minor free graps, classes of graps of bounded expansion and classes wit locally excluded minors. Moreover, wile te latter results relied eavily on grap structure teory, our proof depends on a simple combinatorial property of nowere-dense classes and tus affords a great simplification to te proof. In te sequel, we will use te term efficient algoritm always to mean efficient in te sense of fixed-parameter tractability. Classes on wic efficient algoritms ave previously been obtained for te dominating set problem include planar graps were an algoritm wit running time O(8 k n)-time is given in [2] and graps of genus g, were an O((4g + 40) k n 2 )-time algoritm is given in [14]. Improvements to te algoritms on planar graps ave subsequently been made, to O(4 6 34k n) in [1], to O(2 27 k n) in [18] and to O( k k + n 3 + k 4 ) in [16]. Efficient algoritms for distance-d dominating sets are also known for planar graps and map-graps [10]. For te dominating set problem, efficient algoritms were sown for H-minor free graps in [11]. Te latter generalises te result for classes of graps of bounded genus. More recently, Alon and Gutner gave a linear time parameterized algoritm for dominating sets on d-degenerate graps running in time k O(dk) n [3]. Tis is a furter generalisation beyond H-minor-free classes. Anoter generalisation is obtained by Pilip et al. [23] wo sow tat te problem is fixed-parameter tractable on graps tat exclude K i,j as a subgrap. It sould be noted tat wile all oter classes mentioned above also admit an efficient algoritm for te distance-d dominating-set problem, tis is not te case for classes of degenerate graps. Indeed, tis problem is W[2]-ard, already on te class of 2-degenerate graps [17]. Oter generalisations of H-minor free classes tat ave been considered in te literature are classes wit locally excluded minors [9] and classes of bounded expansion [20]. For te former, it follows from [9] tat te distance-d dominating-set problem is FPT. Tis is because te problem can be specified by a first-order formula (depending on d and k), and any property so specified is FPT on classes tat locally exclude a minor. For classes of bounded expansion, Nešetřil and Ossona de Mendez [22] sow tat te dominating set problem is solvable in fixed-parameter linear time, wile te question of weter te distance-d dominating-set problem is FPT on suc classes is left open. Indeed, tey point out tat teir metod cannot be used to sow tat te distance-2 dominating-set problem is FPT on classes of bounded expansion. Our result settles tis question as it implies te existence of an efficient algoritm for distance-d dominating-set on bounded-expansion classes. Our main results concern classes of graps tat are nowere dense. Tis is a concept introduced by Nešetřil and Ossona de Mendez [19, 21] tat generalises bot locally excluded minors and bounded expansion in te sense tat any class of graps tat eiter locally excludes a minor or as bounded expansion is also nowere dense. Nešetřil and Ossona de Mendez sow tat nowere-dense classes can be caracterised by te property of being uniformly quasi-wide (see Section 2 for te defintions). Te latter is a property introduced by Dawar [7, 8] in te study of preservation teorems in finite model teory. In te present pa- 2

3 per we sow tat tis property is by itself sufficient to establis tat a class of graps admits an efficient parameterized algoritm for distance-d dominating set. Te great advantage ere is tat tis is a combinatorial property tat is easy to state and yields a transparently simple algoritm. Tis sould be contrasted wit te algoritms [10, 11] on H-minor free graps tat eavily rely on grap structure teory. We begin by establising some basic terminology and notation in Section 2, and introduce nowere-dense classes and uniformly quasi-wide classes of graps. In Section 3 we examine te relationsip between tese two notions and extract te algoritmic content of te equivalence between tem. Tis allows us, in Section 4, to exibit an efficient parameterized algoritm for te distance-d dominating set problem on nowere dense classes. In Section 5, we explain ow te same ideas can be carried over to a number of oter parameterized problems tat are defined in terms of domination in graps. 2 Preliminaries For a grap G and vertices u, v V(G), we write dist G (u, v) for te distance (i.e. te lengt of te sortest pat) from u to v. We write Nd G (v) for te d-neigbourood of v, i.e. te set of vertices u in V(G) wit dist G (u, v) d. Were te meaning is clear from context, we drop te superscript G. For positive integers i < j, we write [i, j] for te set {k : i k j}. For a grap G and a set of vertices X V(G), we write G X for G[V(G) \ X], i.e. te subgrap of G induced by te vertices V(G) \ X. DEFINITION 1. Let G be a grap and d N. 1. A set X V(G) is d-scattered if for u = v X, N d (u) N d (v) =. 2. A set X V(G) d-dominates a set W V(G) if for all u W tere is a v X suc tat u N d (v). 3. A set X V(G) is a d-dominating set if it d-dominates V(G). We say tat a grap H is a minor of G (written H G) if H can be obtained from a subgrap of G by contracting edges. An equivalent caracterization (see [12]) states tat H is a minor of G if tere is a map tat associates to eac vertex v of H a non-empty connected subgrap G v of G suc tat G u and G v are disjoint for u = v and wenever tere is an edge between u and v in H tere is an edge in G between some node in G u and some node in G v. Te subgraps G v are called branc sets. We say tat H is a minor at dept r of G (and write H r G) if H is a minor of G and tis is witnessed by a collection of branc sets {G v v V(H)}, eac of wic induces a grap of radius at most r. Tat is, for eac v V(H), tere is a w V(G v ) suc tat G v N G v r (w). Te definition of nowere-dense classes is due to Nešetřil and Ossona de Mendez [19, 21]. We are particularly interested in classes were te excluded minors are computable and for tis purpose we introduce te notion of effectively nowere-dense classes. DEFINITION 2.[nowere dense classes] A class of graps C is said to be nowere dense if for every r 0 tere is a grap H r suc tat H r r G for all G C. We say C is effectively nowere dense if te map r H r is computable. It is immediate from te definition tat if C excludes a minor ten it is nowere dense. It is also not difficult to sow tat classes of bounded expansion and classes tat locally 3

4 exclude minors are also nowere dense. Nešetřil and Ossona de Mendez [19] sow an interesting relationsip between nowere dense classes and a property of classes of structures introduced by Dawar [7, 8] called quasi-wideness. Again, we are interested in effective versions of tis concept. DEFINITION 3.[quasi-wide classes] Let s : N N be a function. A class C of graps is quasi-wide wit margin s if for all r 0 and m 0 tere exists an N 0 suc tat if G C and G > N ten tere is a set S V(G) wit S < s(r) suc tat G S contains an r-scattered set of size at least m. We say tat C is quasi-wide if tere is some s suc tat C is quasi-wide wit margin s. We say tat C is effectively quasi-wide if s and N(r, m) are computable. We occasionally refer to a set S as in te above definition as a bottleneck of G. It turns out tat if C is closed under taking induced subgraps, ten it is nowere dense if, and only if, it is quasi-wide. For suc classes, quasi-wideness is equivalent to te notion of uniform quasi-wideness defined below, wic is te notion we will use in te present paper. DEFINITION 4.[uniformly quasi-wide classes] A class C of graps is uniformly quasi-wide wit margin s if for all r 0 and all m 0 tere exists an N 0 suc tat if G C and W V(G) wit W > N ten tere is a set S V(G) wit S < s(r) suc tat W contains an r-scattered set of size at least m in G S. C is effectively uniformly quasi-wide if s and N(r, m) are computable. We often write s C for te margin of te class C, and N C (r, m) for te value of N it guarantees for tis class. We are only interested in classes C for wic tese functions are computable, and we tacitly make tis assumption from now on. We can now state te equivalence of te two central notions. THEOREM 5.[19] Any class C of graps tat is closed under taking induced subgraps is quasi-wide if, and only if, it is nowere dense. In Section 3, we will exibit te algoritmic content of tis equivalence by sowing tat in any nowere-dense class, tere is an efficient (in te sense of fixed-parameter tractability) algoritm tat can find te bottleneck S and te scattered set induced by its removal. In particular tis implies tat a class C closed under subgraps is effectively nowere dense if, and only if, it is effectively quasi-wide. We end tis section wit some examples of quasiwide classes. EXAMPLE Bounded-degree graps. Te class of graps D d of valence at most d is quasi-wide wit margin 1 and N Dd (r, m) = (d 1) r + d H-minor free graps. Te class of graps excluding H as a minor is quasi-wide wit margin H 1. 3 Computing Bottlenecks and Scattered Sets In tis section, our aim is to extract te computational content of Teorem 5 stating te equivalence between nowere dense classes and uniformly quasi-wide classes. In particu- 4

5 lar, we sow tat in any nowere dense class C, we can efficiently extract bottlenecks and scattered sets in any grap. Te first step is to sow tat in any uniformly quasi-wide class wit margin s, we can compute, from s(r) and N C, a bound on te order of te graps tat are excluded as minors of dept r. LEMMA 7. If C is a uniformly quasi-wide class wit margin s and > N C (r + 1, s(r + 1) + 1), ten K r G for any G C. PROOF. Suppose, for contradiction, tat K r G and let u 1,..., u be suc tat te neigbouroods Nr G (u i ) contain branc sets G 1,..., G witnessing tis. Ten, by te coice of and te definition of quasi-wideness, tere is a set S V(G) wit S < s(r + 1) suc tat {u 1,..., u } contains an r + 1-scattered set A of size s(r + 1) + 1 in G S. Tus, since te branc sets are pairwise disjoint, tere must be two distinct vertices u i, u j A suc tat S G i = S G j =. Tere is an edge between some vertex in G i and some vertex in G j (since tey are branc sets witnessing K r G). We tus ave tat N r+1 (u i ) N r+1 (u j ) = even in G S, contradicting te fact tat A is r + 1-scattered. Te oter direction is based on te following teorem, stated in [8], toug te proof is extracted from tat of a weaker statement proved in [4]. THEOREM 8.[8] For any, r, m 0 tere is an N 0 suc tat if G is a grap wit more tan N vertices ten 1. eiter K r+1 G; or 2. tere is a set S V(G) wit S 2 suc tat G S contains an r-scattered set of size m. Indeed, te bound N is computable as a function of, r and m. To be precise, let R be te function guaranteed by Ramsey s teorem so tat for any set A wit A > R(x, y, z) any colouring of te y-tuples from A wit x distinct colours yields a omogeneous subset of size at least z. Let b (x) = R( + 1,, ( 2)(x + 1)) and let c (x) = R(2, 2, b 2 (x)) were b i (x) denotes te function b iterated i times. Ten, it follows from te construction in [4] tat taking N(, r, m) = c r (m) (i.e. c iterated r times) suffices for te proof of Teorem 8. It follows from te above tat if C is a nowere-dense class of graps wit a computable function suc tat K (r) r G for any G C, ten C is quasi-wide wit computable margin s and a computable function N C. We now sow tat in tis case, we can compute rater more. Tat is, given a grap G C and a set W V(G) wit W > N((r), r, m), we can find, in time O( G 2 ), a set S and a subset A W of at least m elements so tat in G S, A is r-scattered. Tis is formalised in te lemma below, wic relies on extracting te algoritmic content of te proofs in [4]. LEMMA 9. Let C be a nowere-dense class of graps and be te function suc tat K (r) r G for all G C. Te following problem is solvable in time O( G 2 ). Input: G C, r, m N, W V(G) wit W > N((r), r, m) Problem: compute a set S V(G), S (r) 2 and a set A W wit A m, suc tat in G S, A is r-scattered. 5

6 PROOF. In wat follows, we write for (r) and N for N(, r, m). Te proof constructs sequences of sets of vertices W 0 W 1 W r and S 0 S 1 S r = S suc tat for all i, 1. S i < 1 2. W i is i-scattered in G S i 3. c r i (m) < W i 4. for all v S i and u W i tere is a w N G S i i (u) suc tat {v, w} E(G). For i = 0, we take S 0 = and W 0 = W. It is clear tat all four conditions are satisfied. Suppose tat S i and W i ave been constructed. We define a grap G on te set of vertices W i by putting an edge between u and v if tere is an edge in E(G) between some vertex in N G S i i (u) and N G S i i (v). Since K i G, G cannot contain a -clique and tus as W i > c r i I > b 2 (c r (i+1) (m) = R(2, 2, b 2 (m)), wic can be found by a greedy algoritm. Note tat G can be (c r (i+1) (m))), G contains an independent set I wit constructed from G in linear time, tus I is found in quadratic time. Te proof of Lemma 5.2 in [4] ten guarantees tat as long as K i+1 G we can find W i+1 I and S i+1 satisfying te four conditions above. Tis is because te condition K i+1 G guarantees tat tere is a (possibly empty) set of vertices Z in G S i wit S i Z < 1 and a set J I wit J > c r (i+1) (m) suc tat N G S i i+1 (u) N G S i i+1 (v) = Z for eac u, v J. Moreover, te coice of size bounds ensures tat Z can be found by a greedy algoritm. We start by taking Z 0 := and I 0 := I. Once Z j as been constructed (for j < 2), we ceck if tere is a vertex z suc tat tere are more tan b 2 j (c r (i+1) (m)) elements v I j suc tat z N G (S i Z j ) i+1 (v). If tere is, we take I j+1 to be te set of suc elements v and Z j+1 := Z j {z}. Tis process is guaranteed to alt witin at most 2 steps, at wic point a greedy algoritm can find a set J wit at least c r (i+1) (m) vertices wit N G (S i Z j ) i+1 (u) N G (S i Z j ) i+1 (v) =, as oterwise we will ave found K as a minor of G at dept i + 1. Tus, we take S i+1 = S i Z and W i+1 = J to satisfy te four conditions above. Te algoritm for distance-d dominating set we present in Section 4 below makes repeated use of te procedure defined above to recursively reduce te problem of finding a distance-d-dominating set of size k in a grap down to te size N := N C (d, (k + 2)(d + 1) s ), at wic point an exaustive searc is performed. Te running time of te algoritm is tus exponential in N (wic only depends on te parameters), and cubic in G. On te oter and, te exact parameter dependence of te algoritm depends on te function, wic is determined by te class of structures C. However, even for simple classes C, were is linear, or constant, N may be a rater fast-growing function of k and d, as it is defined in terms of iterations of te Ramsey function R. On te oter and, as we saw in Example 6, tere are quasi-wide classes, suc as classes of graps of bounded degree, were N can be bounded by a simple exponential. Te property of being quasi-wide can be seen as stating te existence of weak separators. Recall tat a set S is a separator of a set of vertices W in a grap G if in te grap G S, W is split into non-empty disjoint parts wit no pat between tem. It is known, for instance, tat if G is a grap of treewidt at most, for any set W of vertices, we can find a sep- 6

7 arator S of W wit S + 1. Now, nowere-dense graps are a generalisation of classes of H-minor free graps wic include, in particular, classes of bounded treewidt. Wile we cannot ope for te separator property of te form tat olds on bounded treewidt classes to old in nowere-dense classes, uniform quasi-wideness does sow us tat we can find a small set S tat splits W into parts so tat tere are no sort pats between tem. 4 Distance-d-Dominating Set In tis section, we sow tat te distance-d-dominating set problem is fixed-parameter tractable on any nowere-dense class C of graps, wit parameter d + k. Trougout te remainder of tis section, fix a class C tat is uniformly quasi-wide wit margin s C and let N C (r, m) be as in Definition 4. We consider a more general form of te problem. We are given a grap G and a set W V(G) of vertices and we are asked to determine weter tere is a set X in G of at most k vertices tat d-dominates W. We begin wit te observation tat tis problem, wen parameterized by k, d and te size of W is FPT on te class of all graps. LEMMA 10. Te following problem is fixed parameter tractable. Input: A grap G, W V(G), k, d 0 Parameter: k + d + W Problem: Are tere x 1,..., x k V(G) suc tat W i N d (x i )? PROOF. Consider any partition of W into k sets W 1,..., W k. For eac i [1, k], define te set X i := w W i N d (w). Tat is, X i is te set of vertices tat individually d-dominate te set W i. Now, if eac X i is non-empty, ten we can find te dominating set we are looking for by coosing x i to be any element of X i. Conversely, any set {x 1,..., x k } tat d-dominates W determines a partition W 1,..., W k suc tat x i X i. Te algoritm proceeds by considering eac partition of W into k sets in turn (note tat te number of suc partitions is less tan k W ). For eac partition, we compute te sets X i by computing N d (w) for eac w W and taking appropriate intersections. Tis takes time O(d W G ). Te total running time is terefore O(d W k W G ) Now we want to consider te case were te size of W is not part of te parameter, but G is cosen from te nowere-dense class C. We sow tat in tis case, we can find a set W W wose size is bounded by a function of te parameters k and d suc tat G contains a set of size k tat d-dominates W if, and only if, it contains suc a set tat d-dominates W. Tis will ten allow us to use Lemma 10 to get te desired result. For now, fix k and d, and let s := s C (d) and N := N C (d, (k + 2)(d + 1) s ). Tat is, for any G C and W V(G) wit W > N, we can find S V(G) and A W suc tat S s, A (k + 2)(d + 1) s and A is d-scattered in G S. We claim tat, in tis case, we can efficiently find an element a A suc tat G contains a set of size k tat d-dominates W if, and only if, tere is suc a set tat d-dominates W \ {a}. We formalise tis statement in te lemma below. 7

8 LEMMA 11. Tere is an algoritm, running in time f (k, d) G 2 for some computable function f, tat given G C and W V(G) wit W > N returns a vertex w W suc tat for any set X V(G) wit X k, X d-dominates W if, and only if, X d-dominates W \ {w}. PROOF. By Lemma 9, we can find, in time O( G 2 ), S V(G) and A W suc tat S s, A (k + 2)(d + 1) s and A is d-scattered in G S. Let S = {t 1,..., t s } and, for eac a A, we compute te distance vector v a = (v 1,..., v s ) were v i = dist(a, t i ) if tis distance is at most d and v i = oterwise. Note tat tere are, by construction, at most (d + 1) s distinct distance vectors. Since A (k + 2)(d + 1) s, tere are k + 2 distinct elements a 1,..., a k+2 A wic ave te same distance vector. We claim tat taking w := a 1 satisfies te lemma. CLAIM 12. For any set X V(G) wit X k, X d-dominates W if, and only if, X d- dominates W \ {a 1 }. Te direction from left to rigt is obvious. Now, suppose X d-dominates W \ {a 1 }. Consider te sets A i := N G S d (a i ) for i [2, k + 2]. Tese sets are, by construction, mutually disjoint. Since tere are k + 1 of tem, at least one, say A j, does not contain any element of X. However, since a j W \ {a 1 } tere is a pat of lengt at most d from some element x in X to a j. Tis pat must, terefore, go troug an element of S. Since v a1 = v aj, we conclude tat tere is also a pat of lengt at most d from x to a 1 and terefore X d-dominates W. For te complexity bounds, note tat all te distance vectors can be computed in time O( S A G ). Tis is f (k, d) G for a computable f. Adding tis to te O( G 2 ) time to find S and A gives us te required bound. We now state te main result of tis section. THEOREM 13. Te following is fixed-parameter tractable for any effectively nowere-dense class C of graps. DISTANCE-d-DOMINATING SET Input: A grap G C, W V(G), k, d 0 Parameter: k + d Problem: Determine weter tere is a set X V(G) of k vertices wic d-dominates W. PROOF. Te algoritm proceeds as follows. Compute s := s C (d) and N := N C (d, (k + 2)(d + 1) s ). As long as W > N, use te procedure from Lemma 11 to coose an element w W tat may be removed. Once W N, use te procedure from Lemma 10 to determine weter te required dominating set exists. Tis concludes te proof of Teorem 13. Te following corollaries are immediate. COROLLARY 14. Te dominating set problem is fixed-parameter tractable on any effectively nowere-dense class. Tis generalises te known results for te dominating set problem on classes of bounded expansion and classes tat locally exclude a minor. Tis corollary is also obtained as a consequence of a result in [23]. 8

9 COROLLARY 15. Te distance-d dominating set problem is fixed-parameter tractable wit parameter k + d on any class of graps of bounded expansion, were k is te size of te solution. Tis answers a question of Nešetřil and Ossona de Mendez wo sow tat te dominating set problem is fixed-parameter tractable on suc classes and ask weter te same could be true for te distance-2 dominating set problem. 5 Oter Domination Problems Among problems tat are fixed-parameter intractable, dominating set and its variants play an important role. For instance, in te Compendium of Parameterized Problems [6], a number of problems are given wic are known to be ard in general, but tractable on planar graps. Virtually all of tem are variations on te teme of finding dominating sets. In tis section we sow tat all of tese problems and, in many cases, teir arder distanced versions remain fixed-parameter tractable on nowere-dense classes of graps, wic greatly generalises te results on planar graps. We refer to [6] for formal definitions of te problems below and references to te literature were tey were first studied. Te first type of problems we look at are dominating set problems wit additional requirements for connectivity, suc as CONNECTED DOMINATING SET were we are to compute a dominating set wic induces a connected sub-grap. We study its generalisation to d-domination defined as follows. CONNECTED DISTANCE-d-DOMINATING SET Input: Parameter: Problem: Grap G, k, d N k + d Is tere a subset X V(G) wit X = k suc tat X d-dominates G and G[X] is connected? We are able to sow tat tis problem is FPT on nowere-dense classes of graps by adapting te proof of Lemma 10 to sow tat te following problem is FPT. Input: A grap G, W V(G), k, d 0 Parameter: k + d + W Problem: Are tere x 1,..., x k V(G) suc tat W i N d (x i ) and G[x 1,..., x k ] is connected? Similar metods can be used to sow tat te problem d-connected DISTANCE-d- DOMINATING SET is FPT on nowere-dense classes. Tis is te problem of deciding if tere is a d-dominating set X of k vertices wic is d-connected. A set is said to be d-connected in a grap G if it induces a connected subgrap in te grap G d obtained from G by putting an edge between any two vertices tat ave distance at most d in G. Te same metod sows tat EFFICIENT DOMINATING SET is in FPT on nowere-dense grap classes. EFFICIENT d-dominating SET Input: Parameter: Problem: Grap G, k, d N k + d Is tere a subset X V(G) wit X = k suc tat X is a d-dominating set and, in addition, all pairs x = y X ave distance at least 2d + 1? 9

10 Furter variations of domination problems studied in te literature are ANNOTATED DOM- INATING SET and RED-BLUE DOMINATING SET. Annotated domination means tat we are given a grap G and W V(G) and want to find a set dominating W. Te distance-dversion of tis problem is wat is solved by Teorem 13. Red-Blue Domination means tat we are given a grap G wose vertex set is partitioned into blue and red vertices and we want to dominate te blue vertices using red vertices only. Again its distance-d version can be solved by te metods presented in Section 4. Finally, we look at problems suc as ROMAN DOMINATION, MAXIMUM MINIMAL DOM- INATING SET, PERFECT CODE and DIGRAPH KERNEL. If we are not interested in teir distance-d-version tan an even simpler algoritm tan te one presented above can be used to sow tat tese problems are in FPT on nowere-dense classes of graps, wic we demonstrate using te ROMAN DOMINATION problem. ROMAN DOMINATION Input: Grap G, k N Parameter: k Problem: Is tere a Roman domination function R suc tat v V(G) R(v) k? A Roman domination of G is a function R : V(G) {0, 1, 2} suc tat for all v V(G) if R(v) = 0 ten tere exists an x N(v) suc tat R(x) = 2. To solve te problem on nowere-dense classes of graps we first compute a set S G suc tat G \ S contains a 2- scattered set A of size 2k + 1. Clearly, for at least k + 1 vertices v A we must ave R(v) = 0 and ence one of teir neigbours must be labelled by 2. However, tis implies tat at least one vertex in S must be labelled by 2. As S only depends on te parameter we can use tis to define a recursive procedure wose dept and widt only depend on te parameter. Te following teorem sums up wat we ave establised so far. It is easily seen tat INDEPENDENT SET and INDEPENDENT DOMINATING SET are FPT on nowere-dense classes and our procedures presented before readily solve te problems. We refer to [6] for precise definitions of te problems. THEOREM 16. Te following problems are fixed-parameter tractable on effectively noweredense classes of graps: CONNECTED DOMINATING SET, CONNECTED d-dominating SET, c-connected d-dominating SET, ANNOTATED d-dominating SET, EFFICIENT d- DOMINATING SET, MAXIMUM MINIMAL DOMINATING SET, ROMAN DOMINATION, RED- BLUE DOMINATING SET, INDEPENDENT SET, INDEPENDENT DOMINATING SET, PERFECT CODE, and DIGRAPH KERNEL. Tese examples sow tat te distance-d-dominating set problem tat we sowed is tractable on nowere-dense grap classes is actually representative of a wole class of similar problems wic become tractable in tis case. Several of tese are known to be intractable on grap classes of bounded degeneracy. To give an example of a problem wic remains ard on nowere-dense classes, consider te DIRECTED DISJOINT PATHS problem. DIRECTED DISJOINT PATHS Input: Directed grap G, pairs (s 1, t 1 ),..., (s k, t k ) V(G) 2 Parameter: k Problem: Does G contain k vertex disjoint pats P 1,..., P k suc tat P i links s i and t i? 10

11 Tis is known to be W[1]-ard even on acyclic digraps and it is easy to see tat it can be reduced to te directed disjoint pats problem on graps of degree at most 4 as follows. Let G be a digrap and let v V(G) be a vertex wit in-neigbours u 1,..., u l were l > 1. Let T be a directed rooted tree of degree at most 3 wic as l leaves and were all edges are oriented towards te root. Now eliminate all incoming edges to v and add te tree T to G identifying v wit te root of T and u 1,..., u l wit te leaves of T. A similar procedure is used to eliminate outgoing edges of v. Applying tis to all vertices in G yields a grap G of degree at most 4 but wic as k disjoint pats between te pairs (s 1, t 1 ),..., (s k, t k ) if, and only if, suc pats exist in G. Since te class of graps of degree at most 4 is nowere dense, tis sows te problem is ard on suc classes as in te general case. 6 Conclusion and Furter Work Te aim of te paper is to initiate an algoritmic study of grap classes wic are nowere dense. Te examples above, including te dominating set problem and te more general distance-d dominating set, or (k, d)-centre, problem, demonstrate tat a certain class of important algoritmic problems become fixed-parameter tractable on classes wic are nowere dense. One of te main advantages of tese results over known algoritms for tese problems on classes excluding a fixed minor is tat our algoritms are elementary and do not rely on deep results and metods from grap minor teory. One direction for furter researc is to investigate wat oter problems migt become tractable on nowere-dense classes of graps. Also, it would be interesting to compare nowere-dense classes of graps to grap classes of bounded degeneracy. Te two concepts are incomparable but bot generalise classes excluding a fixed minor and we ave already seen tat tere are examples of problems tat become fixed-parameter tractable on noweredense classes of graps wic are intractable on classes of graps of bounded degeneracy. Finally, it would be interesting to explore te extent of te algoritmic teory of noweredense classes of graps in terms of algoritmic meta-teorems. In particular, it would be very interesting if model-cecking for first-order logic was FPT on nowere-dense classes of graps. Tis would establis a ric algoritmic teory of suc classes. However, establising suc a result would require novel metods as we do not ave a decomposition teory for nowere-dense classes along te lines of wat is used to establis te tractability of first-order logic in classes wit locally excluded minors. References [1] J. Alber, H. L. Bodlaender, H. Fernau, T. Kloks, and R. Niedermeier. Fixed parameter algoritms for dominating set and related problems on planar graps. Algoritmica, 33(4): , [2] J. Alber, H. Fan, M.R. Fellows, H. Fernau, R. Niedermeier, F. Rosamond, and U. Stege. A refined searc tree tecnique for dominating set on planar graps. Journal of Comput. Syst. Sci., 71(4): , [3] N. Alon and S. Gutner. Linear time algoritms for finding a dominating set of fixed size in degenerated graps. In COCOON, pages ,

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