A general kernelization technique for domination and independence problems in sparse classes
AA general kernelization technique for dominationand independence problems in sparse classes
Carl Einarson ∗ Felix Reidl † October 1, 2020
Abstract
We unify and extend previous kernelization techniques in sparse classes [6,17] by defining water lilies and show how they can be used in boundedexpansion classes to construct linear bikernels for ( r, c ) -Dominating Set , ( r, c ) -Scattered Set , Total r -Domination , r -Roman Domination ,and a problem we call ( r, [ λ, µ ]) -Domination (implying a bikernel for r -Perfect Code ). At the cost of slightly changing the output graphclass our bikernels can be turned into kernels.We further demonstrate how these constructions can be combined tocreate ‘multikernels’, meaning graphs that represent kernels for multipleproblems at once. Concretely, we show that r -Dominating Set , Total r -Domination , and r -Roman Domination admit a multikernel; as wellas r -Dominating Set and r -Independent Set for multiple values of r at once. Dominating Set is arguably one of the touchstone for kernelization in sparsegraph classes: after a linear kernel in planar graphs [1] and a polynomial kernel ingraphs defined by an excluded topological minor [2, 12] results for linear kernelsin bounded genus graphs [3] apex-minor-free graphs [9], H -minor-free graphs [10],and finally H -topological-minor-free graphs [11] followed in quick succession. Themost general results to date are linear kernels for bounded expansion classes [6](generalizing all aforementioned classes) and an almost-linear kernels for nowheredense classes [14] (generalizing bounded expansion classes). These latter tworesults even hold for the general problem of r -Dominating Set , where a vertexdominates everything in its closed r -neighbourhood. Together with a recentalmost-linear kernel for the related r -Independence problem [17], these resultsled us to the guiding question: Do the kernelization techniques developed for r -Domination / r -Independence in sparse classes carry over to related problems? Bounded expansion classes.
Nešetřil and Ossona de Mendez introducedbounded expansion classes as a generalization of classes excluding a (topological) ∗ Royal Holloway, University of London, UK [email protected] . † Birkbeck, University of London, UK [email protected] a r X i v : . [ c s . D S ] S e p inor and various useful notions of sparsity ( e . g . embeddability in a surface,bounded degree). In short, a class G has bounded expansion (BE) if any minorobtained by contracting disjoint subgraphs of radius at most r in any member G ∈ G is ∇ r ( G ) -degenerate, where ∇ r ( G ) is a class constant independent of G .There are various equivalent definitions for BE classes [15, 19, 18, 16], all ofwhich have in common that they define families of graph invariants { f r } r ∈ N where r is a parameter governing the ‘depth’ at which the invariant is measured.BE classes then are precisely those graph classes for which f r is finite for everymember of the class. We will not need to work with these invariants directly,instead building on higher-level results discussed in Section 1. Consequently, webroadly refer to these invariants as expansion characteristics . For an in-depthdiscussion see [16]. A selection of problems.
The commonality of the following problems is thatthey can be expressed via universal neighbourhood constraints , meaning that asolution X needs to intersect every ‘neighbourhood’ (a slightly flexible term aswe will see in the following) in at least/at most a certain value.We define an r -dominating set of a graph G to be any set D that satisfies | N r [ u ] ∩ D | (cid:62) for all u ∈ V ( G ) , where N r [ u ] contains all vertices at distance (cid:54) r from u . We arrive at a natural extension of the problem by replacing the righthand side of this domination constraint by an arbitrary constant. We call a setthat satisfies the constraint | N r [ u ] ∩ D | (cid:62) c an ( r, c ) -dominating set and thecorresponding decision problem Input:
A graph G and an integer k . Problem:
Is there a set D ⊆ V ( G ) of size at most k such that | N r [ v ] ∩ D | (cid:62) c for all v ∈ G ? ( r, c ) -Domination parametrised by kFor r = 1 this problem has received some attention in the literature under thename “ k -Domination ” (see e . g . [4]), for c = 1 we recover the above discussedproblem r -Domination .We obtain a slightly different notion of dominance by insisting that verticescannot dominate themselves, but only their neighbourhood. It is natural toextends this notion of total domination by extending the domination radius tosome constant r : Input:
A graph G and an integer k . Problem:
Is there sets D , | D | (cid:54) k such that for every vertex v ∈ G | ( N r [ v ] \ { v } ) ∩ D | (cid:62) ? Total r -Domination parametrised by kFinally, we might think of variants in which domination can occur at differentcost. One such variant is Roman Domination where we can either pay one unitto let a vertex dominate itself (but not its neighbours) or two units to dominate2 vertex and its neighbourhood. We propose the following generalization byallowing domination at distance r : Input:
A graph G , a set L ⊆ V ( G ) and an integer k . Problem:
Is there sets D , D ⊆ V ( G ) with | D | + 2 | D | (cid:54) k such that D r -dominates all of V ( G ) \ D ? r -Roman Domination parametrised by kWhile Roman domination does not quite fit the mould of universal neigh-bourhoods constrains (since we can let vertices ‘opt out’ of the constraint | N r [ v ] ∩ D | (cid:62) ) this deviated is easily encompassed by our kernelizationtechnique.The problem of independence turns out to be closely related to that ofdomination. We define an r -scattered set of a graph G to be any set I thatsatisfies | N r [ u ] ∩ I | (cid:54) for all u ∈ V ( G ) . Note that an r -scattered set isequivalent to a r -independent set (all vertices in I are pairwise at distance > r )and the domination/independence duality that holds in BE-classes (see below)has usually been described with this terminology. However, the natural extensionto ( r, c ) -scattered sets that satisfy the scatter constraints | N r [ u ] ∩ I | (cid:54) c doesnot correspond to independent sets. We therefore opt to speak in terms ofscattered instead of independent sets, in particular, we consider the followingparameterized problem: Input:
A graph G and an integer k . Problem:
Is there a set I ⊆ V ( G ) , | I | (cid:62) k such that | N r [ v ] ∩ I | (cid:54) c forall v ∈ V ( G ) ? ( r, c ) -Scattered Set parametrised by kFinally, we consider the problem that arises when combining the domination-and scatter-constraints into the form λ (cid:54) | N r [ u ] ∩ D | (cid:54) µ , which leads to thefollowing, rather general, parameterized problem: Input:
A graph G and an integer k . Problem:
Is there a set D ⊆ V ( G ) of size at most k such that everyvertex v ∈ G satisfies λ (cid:54) | N r [ v ] ∩ D | (cid:54) µ ? ( r, [ λ, µ ]) -Domination parametrised by kHere ( r, [ c, ∞ ]) -Domination is equivalent to ( r, c ) -Dominating Set and ( r, [0 , c ]) -Domination to ( r, c ) -Scattered Set . The problem further covers well-established problems like Perfect Code which we again generalize by in-troducing a distance-parameter: 3 nput:
A graph G an integer k . Problem:
Is there a set I ⊆ V ( G ) of size at most k such that | N r [ v ] ∩ I | = 1 for all v ∈ V ( G ) ? r -Perfect Code parametrised by k Kernelization in sparse classes.
The definition of a kernel (see [5] for aproblem restricted to a certain input class demands that the output belongs tothis class as well, e . g . a planar kernelization needs to output a planar graph.This turns out to be too restrictive for very general notions of sparseness and weare left with the choice of either outputting an annotated instance belonging toa different problem, called a bikernel , or to modify the graph to ‘simulate’ theannotation in the original problem, but these modifications take the instanceout of the original graph class. Here we settle for the following compromise: aparametrised graph problem P ⊆ G × N for a BE-class G admits a BE kernel ifthere is a kernelization that outputs an instance in G (cid:48) × N with ∇ r ( G (cid:48) ) (cid:54) g ( ∇ r ( G )) for some function g and all r ∈ N . This is justified by the idea that all nicealgorithmic properties stemming from G being BE carry over from G to G (cid:48) withonly changes to some constants —if other properties of the class are of primaryinterest (embedding in a surface, excluded minors, etc .) then the BE-view issimply too coarse. Our results.
Inspired by the kernelization for r -Dominating Set [6] and r -Independent Set [17] in sparse classes, we unify and extend these techniques bydefining a structure we call water lilies and show how their existence can be usedto find small cores , that is, subset of vertices that either are guaranteed to containa solution ( solution core ) or that already fully represent the neighbourhood-constraints governing the problem ( constraint core ). We define and prove the exis-tence of water lilies in BE-classes in Section 3, building on our proof of a constant-factor approximation for ( r, c ) -Dominating Set in BE-classes from Section 2.In Section 4 we use water lilies to prove linear bikernels for ( r, c ) -DominatingSet , ( r, c ) -Scattered Set , Total r -Domination , r -Roman Domination ,and ( r, [ λ, µ ]) -Domination (implying a bikernel for r -Perfect Code ) intoappropriate annotated variants of these problems. We then show in Section 5how these bikernels can be turned into BE-kernels for ( r, c ) -Dominating Set , ( r, c ) -Scattered Set , Total r -Domination , r -Roman Domination , and r -Perfect Code . Finally, in Section 6, we demonstrate how these constructionscan be combined to create ‘multikernels’, meaning graphs that represent kernelsfor multiple problems at once. Concretely, we show that r -Dominating Set , Total r -Domination , and r -Roman Domination admit a multikernel; aswell as r -Dominating Set and r -Independent Set (even for multiple valuesof r at once). Using this argument, we might as well allow a BE-kernel to change the depth as well, i . e . ∇ r ( G (cid:48) ) (cid:54) g ( ∇ g ( r ) ( G )) . In this work we do not need this level of generality and stick to asimpler definition. Notation and previous results
For a maximization problem P defined via universal neighbourhood constraintsand a graph G we call a set L ⊆ V ( G ) a constraint core if for every set D ⊆ V ( G ) it holds that D is a solution to P in G already if the constraints only hold forvertices in L . Analogous, for a minimization problem P defined via universalneighbourhood constraints, we call a set U ⊆ V ( G ) a solution core if a minimumsolution to P already exists inside U . In both cases, note that V ( G ) is always atrivial core and that a superset of any core is a core as well.A set D ⊆ V ( G ) is an ( r, c ) -dominating set if for every vertex v ∈ V ( G ) it holds that | N r [ v ] ∩ D | (cid:62) c . Importantly, this constraint must also hold forvertices contained in D , therefore such a set can only exist if | N r [ v ] | (cid:62) c for all v ∈ G . We write dom cr ( G ) to denote the size of a minimum ( r, c ) -dominatingset in G and let dom cr ( G ) = ∞ if no such set exists.A set I ⊆ V ( G ) is r -independent if every pair of vertices u, v ∈ I hasdistance at least r + 1 . We write ind r ( G ) to denote the size of a maximum r -independent set in G . Related, a set I ⊆ V ( G ) is an ( r, c ) -scattered set iffor all vertices v ∈ G it holds that | N r [ v ] ∩ I | (cid:54) c . An ( r, -scattered set isequivalent to a r -independent set, but this relationship breaks down for c > .We defined sct cr ( G ) as the size of a maximum ( r, c ) -scattered set in G . In allcases, for c = 1 we will omit the superscript.In many of the following constructions we will use the phrase “connect u to v bya path of length r ”. This operation is to be understood as adding ( r − new ver-tices a , . . . , a r − the graph and then adding edges to create the path ua . . . a r − v . We adapted the following results to use the notation introduced above for thesake of a unified presentation. In particular, we will be using sct r insteadof ind r . The function wcol r is one of the expansion characteristics mentionedabove (see e . g . [19] for a definition), here it is enough to know that for everymember G of a BE-class, wcol r ( G ) is bounded by a constant for every r ∈ N . Theorem 1 (Dvořák [7]) . For every graph G and integer r ∈ N it holds that sct r ( G ) (cid:54) dom r ( G ) (cid:54) wcol r ( G ) sct r ( G ) Dvořák recently showed an improved bound [8], we will use the above simpler ex-pression. In the same paper he further proved the following relationship between r -scattered sets and ( r, c ) -scattered sets (translated into our terminology): Theorem 2 (Dvořák [8]) . For every graph G and integers c, r ∈ N it holdsthat c wcol r ( G ) sct cr ( G ) (cid:54) sct r ( G ) (cid:54) sct cr ( G ) Theorem 3 (Dvořák’s algorithm [7]) . For every BE class G and r ∈ N thereexists a constant c dvrk r and a polynomial-time algorithm that computes an r -dominating set D of G and an r -scattered set A ⊆ D with | D | (cid:54) c dvrk r | A | .5n particular, the r -scattered set A witnesses that D is indeed a c dvrk r -approximationof a minimum r -dominating of G . This algorithm can further be modified tocompute a dominating set for a specific set X ⊆ V ( G ) only; in that case itoutputs the sets A and D , A ⊆ D ∩ X , where D dominates all of X in G and A is r -scattered in G . We will call this algorithm the warm-start variant since weonly need to mark the vertices V ( G ) \ X as already dominated and then run theoriginal algorithm (an alternative is a small gadget construction [6]). Given a vertex set X ⊆ V ( G ) we call a path X -avoiding if its internal verticesare not contained in X . A shortest X -avoiding path between vertices x, y isshortest among all X -avoiding paths between x and y . Definition 1 ( r -projection) . For a vertex set X ⊆ V ( G ) and a vertex u (cid:54)∈ X we define the r -projection of u onto X as the set P rX ( u ) := { v ∈ X | there exists an X -avoiding u - v -path of length (cid:54) r } Note in particular that P X ( u ) = N ( u ) ∩ X , but for r > the sets P rX ( u ) and N r ( u ) ∩ X might differ. Definition 2 ( r -shadow) . For a vertex set X ⊆ V ( G ) and a vertex u (cid:54)∈ X wedefine the r -shadow of u onto X as the set S rX ( u ) := { v ∈ V ( G ) | every u - v -path of length (cid:54) r has an internal vertex in X } The shadow S rX ( u ) contains precisely those vertices that are ‘cut off’ by theset P rX ( u ) . We will frequently need the union of shadow and projection andtherefore introduce the shorthand SP rX ( u ) := S rX ( u ) ∪ P rX ( u ) .Two vertices that have the same r -projection onto X do not, however, neces-sarily have the same shadow since the precise distance at which the projectionlies might differ. To distinguish such cases, it is useful to consider the projectionprofile of a vertex to its projection: Definition 3 ( r -projection profile) . For a vertex set X ⊆ V ( G ) and a vertex u (cid:54)∈ X we define the r -projection profile of u wrt X as a function π rG,X [ u ] : X → [ r ] ∪ ∞ where π rG,X [ u ]( v ) for v ∈ X is the length of a shortest X -avoiding pathfrom u to v if such a path of length at most r exists and ∞ otherwise.We say that a function ν : X → [ r ] ∪ ∞ is realized on X (as a projection profile)if there exists a vertex u (cid:54)∈ X for which ν = π rG,X [ u ] and we denote the set of allrealized profiles by Π rG ( X ) . We will usually drop the subscript G if the graph isclear from the context. It will be convenient to define an equivalence relationthat groups vertices outside of X by their projection profile. Define u ∼ rX v ⇐⇒ π rX [ u ] = π rX [ v ] for pairs u, v ∈ V ( G ) \ X .It turns out that in BE classes, the number of possible projection profilesrealised on a set X is bounded linearly in the size of X .6 emma 1 (Adapted from [6, 14]) . For every BE class G and r ∈ N thereexists a constant c proj r such that for every G ∈ G and X ⊆ V ( G ) , the number of r -projection profiles realised on X is at most c proj r | X | .In our notation this can alternatively be written as | Π r ( X ) | = | ( V ( G ) \ X ) / ∼ rX | (cid:54) c proj r | X | . We will crucially rely on the following two results for BE classes: Lemma 2 (Projection closure [6]) . For every BE class G and r ∈ N thereexists a constant c projcl r and a polynomial-time algorithm that, given G ∈ G and X ⊆ V ( G ) , computes a superset X (cid:48) ⊇ X , | X (cid:48) | (cid:54) c projcl r | X | , such that | P rX (cid:48) ( u ) | (cid:54) c projcl r for all u ∈ V ( G ) \ X (cid:48) . Lemma 3 (Shortest path closure [6]) . For every BE class G and r ∈ N thereexists a constant c pathcl r and a polynomial-time algorithm that, given G ∈ G and X ⊆ V ( G ) , computes a superset X (cid:48) ⊇ X , | X (cid:48) | (cid:54) c pathcl r | X | , such that for all u, v ∈ X with dist ( u, v ) (cid:54) r it holds that dist G [ X (cid:48) ] ( u, v ) = dist ( u, v ) .It will be useful to combine the above two lemmas in the following way: Definition 4 (Projection kernel) . Given a graph G and a set X ⊆ V ( G ) , an ( r, c ) -projection kernel of ( G, X ) is an induced subgraph ˆ G of G with X ⊆ V ( ˆ G )and the following properties:1. N d ˆ G ( v ) ∩ X = N dG ( v ) ∩ X for all v ∈ X and d (cid:54) r ; and2. if the signature ν : X → [ r ] ∪ ∞ is realized on X by p distinct verticesin G , then ν is realized by at least min { c, p } distinct vertices in ˆ G . Lemma 4.
For every BE class G and c, r ∈ N there exists a constant c total r,c anda polynomial-time algorithm that, given G ∈ G and X ⊆ V ( G ) , computes an ( r, c ) -projection kernel ˆ G of ( G, X ) with | ˆ G | (cid:54) c total r,c | X | . Proof.
We first apply Lemma 2 to X and obtain a set X ⊃ X , | X | (cid:54) c projcl r | X | ,such that the projections of outside vertices onto X have size at most c projcl r .Next, we apply Lemma 3 to X and receive a set X ⊃ X , | X | (cid:54) c pathcl r | X | ,such that the graph G [ X ] preserves short distances (less than or equal to r )between vertices in X . Finally, let U contain up to c representatives for everyequivalence class [ u ] ∈ V ( G ) / ∼ rX (if the class is smaller than c we include allof it). By Lemma 1 we have that | U | (cid:54) c · c proj r | X | .Construct now X by taking the union X ∪ U as well as shortest paths fromevery member u ∈ X ∪ U to all of P rX ( u ) . By definition, each of these paths haslength at most r and therefore contains at most r − internal vertices. Since, byconstruction of X , | P rX ( u ) | (cid:54) c proj r ; it follows that we add at most c proj r ( r − vertices per vertex in X ∪ U . Taking the above bounds together, we have that | X | (cid:54) ( r − c proj r ( c pathcl r + c · c proj r ) | X | =: c total r,c | X | . It remains to be shown that ˆ G := G [ X ] has the desired properties.Property 1 follows directly from the fact that already G [ X ] ⊆ ˆ G preservesshort distances among vertices inside X ⊇ X . In particular, each vertex in X \ X has the same r -projection profile onto X in G and ˆ G .7o see that Property 2 holds, consider any profile ν realized on X by ver-tices S ⊆ V ( G ) \ X in G . First consider the case S \ X (cid:54) = ∅ . Then by construction,the set U contains min { c, | S \ X |} vertices from S \ X that realize ν in G andwhose projection onto X is the same in G and ˆ G . Since X ⊇ X , we concludethat their projection on X in ˆ G must be ν . By the above, the vertices in S ∩ X must have the profile ν as well. Now assume S ⊆ X , therefore no vertex outsideof X has the profile ν in G . As argued above, S has the profile ν in ˆ G as well,therefore ˆ G contains | S | (cid:62) min { c, | S |} vertices with profile ν , as claimed.Note that the above construction implies that Π r ˆ G ( X ) ⊇ Π rG ( X ) , however, it isnot necessarily true that Π r ˆ G ( X ) = Π rG ( X ) .The following is a slight restatement of Theorem 4 in [13]. We emphasisethat the proof by Kreutzer et al . is actually constructive and can be implementedto run in polynomial time. Lemma 5 (UQW in BE classes [13]) . For every BE class G and distance d ∈ N there exists a constant c UQW d and a polynomial-time algorithm that, given G ∈ G ,a size t ∈ N and X ⊆ V ( G ) with | X | (cid:62) c UQW d · t , computes a set S of size atmost ( c UQW d ) and X (cid:48) ⊆ X \ S of size at least t such that X (cid:48) is d -scattered in G − S . ( r, c ) -Dominating Set Theorem 4.
Let G be a BE class and fix r, c ∈ N . There exists a constant c cdom r,c and an algorithm that, for every G ∈ G , computes in polynomial time an ( r, c ) -dominating set of size at most c cdom r,c dom cr ( G ) or concludes correctly that G cannot be ( r, c )-dominated. Proof.
We compute a sequence of dominating sets D , D , . . . , D c with theinvariants that a) D i ( r, i ) -dominates G and b) | D i +1 | (cid:54) c dvrk r c proj r | D i | + c dvrk r dom i +1 r ( G ) .To start the process, let D be an c dvrk r -approximate r -dominating set for G ,this set clearly satisfies invariant a). We proceed in two steps to construct D i +1 from D i . Build the set U i as follows: for every projection µ ∈ Π r ( D i ) realizedby an equivalence class [ v ] ∈ ( V ( G ) \ D i ) / ∼ rD i we pick one (arbitrary) vertexfrom S rD i ( v ) \ D i and add it to U i , if such a vertex exists. Then for every vertex u ∈ D i that is not ( i + 1) -dominated by D i ∪ U i , we add an arbitrary vertexfrom N r [ u ] \ D i to U i (note that if no such vertex exists we conclude that G cannot be ( r, c ) -dominated).By construction, the size of U i is bounded by | U i | (cid:54) | Π r ( D i ) | + | D i | (cid:54) ( c proj r + 1) | D i | . Further note that every vertex in D i ∪ U i is ( r, i + 1) -dominatedby D i ∪ U i : due to invariant a), the set D i ( r, i ) -dominates D i ∪ U i and U i nowadditionally dominates itself (at least) once and, by construction, those verticesin D i that are not yet ( r, i + 1) -dominated by D i .Define the set R i to contain all vertices that are not ( r, i + 1) -dominated by D i ∪ U i , note that in particular N r [ R i ] ∩ U i = ∅ . Let G (cid:48) = G − ( D i ∪ U i ) . Apply8vořák’s warm-start algorithm to find a distance- r dominator D (cid:48) i for R i in G (cid:48) and a r -scattered set A (cid:48) i ⊆ D (cid:48) i ∩ R i with | A (cid:48) i | (cid:54) | D (cid:48) i | (cid:54) c dvrk r | A (cid:48) i | . Claim. | A (cid:48) i | (cid:54) ( c proj r + 1) | D i | + dom i +1 r ( G ) . Proof.
Let X be an ( r, i + 1) -dominating set of G of minimum size and assumethat | A (cid:48) i | > ( c proj r + 1) | D i | + dom i +1 r ( G ) (cid:62) | U i ∪ X | . Then there exists a ∈ A (cid:48) i such that N rG (cid:48) [ a ] ∩ ( U i ∪ X ) = ∅ . Since X ( r, i + 1) -dominates a but D i ∪ U i doesnot (because a ∈ R i ) there must be at least one vertex b ∈ X ∩ ( N rG [ a ] \ N rG (cid:48) [ a ]) that is not contained in D i ∪ U i . This means that b ∈ S rD i ∪ U i ( a ) and since N rG [ a ] ∩ U i = ∅ , we have that S rD i ∪ U i ( a ) = S rD i ( a ) and therefore even b ∈ S rD i ( a ) .But then, since b (cid:54)∈ D i ∪ U i , we could have added b to U i during the firstconstruction phase in order to dominate the class [ a ] . The existence of a leads usto a contradiction and we conclude that | A (cid:48) i | (cid:54) ( c proj r + 1) | D i | + dom i +1 r ( G ) .Finally, construct the set D i +1 = D (cid:48) i ∪ D i ∪ U i . Since D (cid:48) i r -dominates R i which,by construction, were the only vertices not yet ( r, i + 1) -dominated by D i ∪ U i ,we conclude that D i +1 is indeed an ( r, i + 1) -dominating set of G ; thus invarianta) is preserved. To see that invariant b) holds, let us bound the size of D i +1 : | D i +1 | (cid:54) | D (cid:48) i | + | D i | + | U i | (cid:54) c dvrk r | A (cid:48) i | + | D i | + ( c proj r + 1) | D i | (cid:54) c dvrk r ( c proj r + 1) | D i | + c dvrk r dom i +1 r ( G ) + ( c proj r + 2) | D i | = (cid:0) ( c dvrk r + 1)( c proj r + 1) + 1 (cid:1) | D i | + c dvrk r dom i +1 r ( G ) (cid:54) (cid:0) c dvrk r c proj r + 1 (cid:1) | D i | + c dvrk r dom i +1 r ( G ) (cid:54) c dvrk r c proj r | D i | + c dvrk r dom i +1 r ( G ) . We conclude that invariant b) holds, as claimed. Resolving the recurrenceprovided by this inequality, we finally obtain the bound | D c | (cid:54) c dvrk r c (cid:88) i =1 (5 c dvrk r c proj r ) c − i dom ir ( G ) (cid:54) (5 c dvrk r c proj r ) c +1 dom cr ( G ) , and the claim follows with c cdom r,c := (5 c dvrk r c proj r ) c +1 . Definition 5 (Water lily) . A water lily of radius r , depth d (cid:54) r and adhesion c in a graph G is a tuple ( R, C ) of disjoint vertex sets with the following properties: • C is r -scattered in G − R , • N rG − R [ C ] is ( d, c ) -dominated by R in G .We call R the roots , C the centres , and the sets { N rG − R [ x ] } x ∈ C the pads of thewater lily.A water lily is uniform if all centres have the same d -projection onto R , e . g . π dR [ x ] is the same function for all x ∈ C . The ratio of a water lily is anyguaranteed lower bound on | C | / | R | . 9he following lemma lies at the heart of our unification of previous techniques[6, 14, 17]. It streamlines the construction of BE-kernels considerably, as we willsee in the following section. Lemma 6.
For every BE class G and c, r, d ∈ N , d (cid:54) r , there exist constants c scale c,r,d , c margin c,r,d , c base r,d with the following property: for every G ∈ G which hasan ( r, c ) -dominating set, t ∈ N and A ⊆ V ( G ) with | A | (cid:62) c scale c,r,d · ( c base r,d ) t · dom cd ( G ) there exists a uniform water lily ( R, C ) , C ⊆ A , with depth d , radius r ,adhesions c and with | R | (cid:54) c margin c,r,d , | C | (cid:62) t . Moreover, such a water lily can becomputed in polynomial time. Proof.
Given G , we use Theorem 4 to compute a ( d, c ) -dominating set D (cid:48) ofsize at most c cdom r,c · dom d ( G ) in polynomial time or conclude that no suchset exits. Afterwards, we compute the ( r + d ) -projection closure D of D (cid:48) , byLemma 2 we have that | D | (cid:54) c projcl r + d | D (cid:48) | and thus | D | (cid:54) c projcl r + d c cdom r,c dom d ( G ) .Let A (cid:48)(cid:48) := A \ D , we will later choose c scale c,r,d so that A (cid:48)(cid:48) is still large enough forthe following arguments to go through.Define the equivalence relation ∼ D over A (cid:48)(cid:48) via a ∼ D a (cid:48) ⇐⇒ π r + dD [ a ] = π r + dD [ a (cid:48) ] . By Lemma 1, the number of classes in A (cid:48)(cid:48) / ∼ D is bounded by c proj r + d | D | ;by an averaging argument we have at least one class [ a ] ∈ A (cid:48)(cid:48) / ∼ D of size (cid:12)(cid:12) [ a ] (cid:12)(cid:12) (cid:62) | A (cid:48)(cid:48) | c proj r + d | D | (cid:62) | A | − | D | c proj r + d | D | . Let R (cid:48)(cid:48)(cid:48) be P r + dD ( a ) , i . e . the ( r + d ) -projection of [ a ] ’s members on D . By ourearlier application of Lemma 2 we have that | R (cid:48)(cid:48)(cid:48) | = | P r + dD ( a ) | (cid:54) c projcl r + d .Again, we will choose c scale c,r,d large enough to apply Lemma 5 with distance r and size c proj d | R (cid:48)(cid:48)(cid:48) | t to the set [ a ] and receive a subset A (cid:48) ⊆ [ a ] of size at least c proj d ( c UQW r + c projcl r + d ) · t and a set R (cid:48)(cid:48) ⊆ V ( G ) \ A (cid:48) , | R (cid:48)(cid:48) | (cid:54) c UQW r , such that A (cid:48) is r -scattered in G − R (cid:48)(cid:48) . Let R (cid:48) := R (cid:48)(cid:48) ∪ R (cid:48)(cid:48)(cid:48) , by the above bounds on R (cid:48)(cid:48) and R (cid:48)(cid:48)(cid:48) it follows that | R (cid:48) | (cid:54) c UQW r + c projcl r + d . By Lemma 1 and the fact that | A (cid:48) | (cid:62) c proj d ( c UQW r + c projcl r + d ) · t (cid:62) | Π d ( R (cid:48) ) | · t Figure 1: Schematic of a water lily ( R, C ) with radius r , depth d and adhesion c .Removing the ‘tangled’ roots R creates disjoint r -neighbourhoods around C which we imagine like lily pads floating on a pond.10here exists a set C ⊆ A (cid:48) of size at least t such that all members of C have thesame d -projection onto R (cid:48) .We construct the set R from R (cid:48) as follows: for every projection profile µ ∈ Π d ( R (cid:48) ) realized by a class [ u ] ∈ N rG − R (cid:48) [ C ] / ∼ dR (cid:48) we add max { , c − | P dR (cid:48) ( u ) |} vertices from the shadow S dR (cid:48) ( u ) ∩ D (cid:48) . Since D (cid:48) ( d, c ) -dominates all of G , suchvertices must exist. By construction, | R | (cid:54) c | R (cid:48) | and R ( c, d ) -dominates all of N rG − R (cid:48) [ C ] and thus in particular N rG − R [ A (cid:48) ] . Note further that all vertices weadded lie inside S r + dR (cid:48) [ C ] , therefore the projection profiles of C are not changed bythis operation (all paths of length at most r + d from C to vertices in R/R (cid:48) passthrough R (cid:48) ). We conclude that the uniformity condition holds on ( R, C ) . Thisconstruction also provides us with the bound | R | (cid:54) c ( c UQW r + c projcl r + d ) =: c margin c,r,d .Finally, let us determine a value for c scale c,r,d that suffices for the above con-struction to go through. In order to apply Lemma 5, we need that | [ a ] | (cid:62) c UQW r · c proj d ( c UQW r + c projcl r + d ) · t , accordingly we need that | A | − | D | c proj r + d | D | (cid:62) c UQW r · c proj d ( c UQW r + c projcl r + d ) · t ⇐ = | A | c proj r + d | D | (cid:62) c UQW r · c proj d ( c UQW r + c projcl r + d ) · t ⇐ = | A | c proj r + d c projcl r + d c cdom r,c dom cd ( G ) (cid:62) c UQW r · c proj d ( c UQW r + c projcl r + d ) · t We conclude that choosing the constants c scale c,r,d = 2 c proj r + d c projcl r + d c cdom r,c c UQW r and c base r,d = 2 c proj d ( c UQW r + c projcl r + d ) suffices to prove the claim.We can impose even more structure on a water lily in the following sense: let usdefine a pad signature as a function σ : C → Σ ∗ (for some alphabet Σ ) that canbe computed by a polynomial-time algorithm receiving the following inputs: • The depth d , radius r and adhesion c of the water lily; • the centre a , its pad N rG − R [ a ] , the roots R ; • the subgraph G [ R ∪ N rG − R [ a ]] alongside potential vertex/edge labels fromthe host graph G .We say that σ is bounded if the size of its image can be bounded by a constant.Every pad signature σ gives rise to an equivalence relation ∼ σ ⊆ C × C for awater lily ( R, C ) via a ∼ σ a (cid:48) ⇐⇒ σ ( a ) = σ ( a (cid:48) ) . Note that if σ is bounded, then ∼ σ has finite index. A water lily is σ -uniform ifall its centres belong to the same equivalence class under ∼ σ ; or alternativelyif all centres have the same image under σ . For a bounded signature σ , wefind a ∼ σ -uniform water lily of ratio τ by first finding a water lily ( R (cid:48) , C (cid:48) ) withratio p · τ , where p is an upper bound on the image of σ , and then return R (cid:48) together with the largest class in C (cid:48) / ∼ σ . Accordingly:11 orollary 1. For every BE class G , c, r, τ ∈ N and pad signature σ with finiteindex there exists a constant c lily = c lily c, r,r,τ,σ with the following property: forevery G ∈ G which has an ( r, c ) -dominating set and A ⊆ V ( G ) with | A | (cid:62) c lily · dom cd ( G ) there exists a σ -uniform water lily ( R, C ) , C ⊆ A , | R | (cid:54) c lily , ofdepth r , radius r , adhesion c and ratio τ .Moreover, such a water lily can be computed in polynomial time.Let us define a particular bounded pad signature that will be useful in theremainder: define ν as ν ( a ) := ( { π dR [ x ] | x ∈ N iG − R ( a ) } | (cid:54) i (cid:54) r (cid:1) , where the right-hand side is to be understood as encoded in a string by somesuitable scheme. Two centres are equivalent under ∼ ν if they have the sameprojection-types at the same distance (though potentially at different multi-plicities) inside their respective pads. Since | R | has constant size according toLemma 6 and there are at most c proj d | R | possible projection profiles according toLemma 1, the image of ν has size at most r c proj d | R | (cid:54) r c proj d c lily and therefore ν isa bounded pad signature.We will sometimes combine ν with a finite number of vertex labels that ariseduring the construction of bikernels. If vertices are labelled by f : V ( G ) → Σ for some finite alphabet Σ , then we understand ν to be the above equivalencerelation further refined by the equivalence relation u ∼ f v ⇐⇒ f ( u ) = f ( v ) . We show in the following that a range of problems over hereditary BE-classesadmit linear bikernels in the same class (see the full version for r -RomanDomination and Total r -Domination ). The target problem in all three casesis a suitable annotated version of the original problem, which we define justahead of each proof. Input:
A graph G , a set L ⊆ V ( G ) and an integer k . Problem:
Is there a set D ⊆ V ( G ) of size at most k such that | N r [ v ] ∩ D | (cid:62) c for all v ∈ L ? Annotated ( r, c ) -Domination parametrised by k Theorem 5. ( r, c ) -Dominating Set over a hereditary BE-class G admits alinear bikernel into Annotated ( r, c ) -Dominating Set over the same class G .Moreover, the resulting graph is an ( r, c ) -projection kernel of the original graph. Proof.
Let ( G, k ) be an input where G is taken from a BE class. As a first step,we deal with the case dom µr ( G ) large by computing an ( r, µ ) -dominating setusing the algorithm from Theorem 4. If it returns a solution larger than c cdom r,c k ,we conclude that dom µr ( G ) > k in which case we return a trivial no-instance.12therwise, we show that ( r, c ) -Dominating Set admits a linear constraint coreand then show how to construct a BE-kernel from that core.Otherwise we now show that ( r, c ) -Dominating Set admits a linear con-straint core and then show how to construct a BE-kernel from that core. Claim. ( r, c ) -Dominating Set has a linear constraint core in BE classes. Proof.
Let L ⊆ V ( G ) be a constraint core of G with | L | (cid:62) c lily c, r,r, dom cr ( G ) .By Corollary 1, we can find in polynomial time a uniform water lily ( R, C ) , C ⊆ L , | R | (cid:54) c lily of depth r , radius r , adhesion c and ratio . Let a ∈ C bean arbitrary centre, we claim that L \ { a } is still a constraint core, that is, everyset that ( r, c ) -dominates L \ { a } will also ( r, c ) -dominate a .To that end, let D be a minimum ( r, c ) -dominating set and define D (cid:48) := D \ N rG − R [ C ] . If D (cid:48) ( r, c ) -dominates any part of C , it dominates all of C (andtherefore a ) as ( R, C ) is uniform. Thus assume that D (cid:48) does not ( r, c ) -dominate C . Consider the case where a set S ⊆ D ∩ N rG − R [ C ] exists such that everyvertex in S dominates more than one vertex in C . If | S | (cid:62) c then S alonealready ( r, c ) -dominates all of C and thus in particular a . In all remaining cases,every set N rG − R [ a (cid:48) ] , a (cid:48) ∈ C must contain at least one vertex from D and weconclude that | D \ D (cid:48) | (cid:62) | C | (cid:62) | R | . Let ˜ D := D (cid:48) ∪ R , we claim that ˜ D isan ( r, c ) -dominating set of G . Simply note that the only vertices that are not ( r, c ) -dominated by D (cid:48) lie inside N rG − R [ C ] —but this is precisely the set that is ( r, c ) -dominated by R . We arrive at a contradiction since | D | = | D \ D (cid:48) | + | D (cid:48) | (cid:62) | R | + | D (cid:48) | > | R | + | D (cid:48) | (cid:62) | ˜ D | and we assumed D to be minimum. Thus L \ { a } is a constraint core for ( r, c ) -Dominating Set in G . We iterate this procedure until | L | < c lily c, r,r, dom cr ( G ) and end up with a linear constraint core.In the following, let L ⊆ V ( G ) be a constraint core for ( G, k ) with | L | (cid:54) c lily dom cr ( G ) and let O = V ( G ) \ L . If | L | > c lily k , we can conclude that k > dom cr ( G ) and output a trivial no-instance, thus assume from now on that | L | (cid:54) c lily k .We apply Lemma 4 with X = L and r , c as here to obtain a projection kernel ˆ G with | ˆ G | (cid:54) c total r,c | L | = O ( k ) which a) preserves (cid:54) r -neighbourhoods in L and b)realizes every r -projection onto L that is realized p times in G at least min { c, p } times. We claim that ( G, k ) is equivalent to the annotated instance ( ˆ G, L, k ) .Assume that D is an ( r, c ) -dominating set of G , clearly it is also a solution tothe annotated instance ( G, L, k ) . Partition D into D L = D ∩ L and D O = D \ L .Consider x ∈ D O and note that | [ x ] ∩ D O | < c for the r -neighbourhoodclass [ x ] ∈ O/ ∼ rL since otherwise we could remove a vertex from [ x ] ∩ D O from D and still ( r, c ) -dominate all of L . With this observation, construct the set ˆ D O asfollows: for every vertex x ∈ D O we include | [ x ] ∩ D O | vertices from O ∩ V ( ˆ G ) in ˆ D O , by property b) of the projection kernel ˆ G we know that at least c suchvertices are available. Then the set ˆ D := D L ∪ ˆ D O ( r, c ) -dominates all of L in ˆ G ,13y property a) of ˆ G , and we are done. In the other direction, let ˆ D be an ( r, c ) -dominator of L in ˆ G . By property a) and b) of ˆ G the set ˆ D therefore also ( r, c ) -dominates L in G , and since L is a constraint core of G it then ( r, c ) -dominatesall of G . We conclude that ( ˆ G, L, k ) is equivalent to ( G, k ) and | ˆ G | = O ( k ) . Input:
A graph G , a set L ⊆ V ( G ) and an integer k . Problem:
Is there sets D , | D | (cid:54) k such that for every vertex v ∈ L | ( N r [ v ] \ { v } ) ∩ D | (cid:62) ? Annotated Total r -Domination parametrised by k Theorem 6.
Total r -Domination over a hereditary BE-class G admits alinear bikernel into Annotated Total r -Domination over the same class G .Moreover, the resulting graph is a ( r, -projection kernel of the original graph. Proof.
Every r -total dominating set is in particular an r -dominating set and, onthe other hand, we can turn an r -dominating set D into an r -dominating set ofsize at most | D | by including at most one neighbour for each vertex in D .Hence, given an input ( G, k ) to Total r -Domination with G taken from aBE class, we verify as a first step that dom r ( G ) is not too large by computing an r -dominating set using the algorithm described in Theorem 4. If the algorithmreturns a solution larger than c dvrk r k , we conclude that dom r ( G ) > k andtherefore G does not have a r -total dominating set of size k . In this case we outputa trivial no-instance, thus assume for the remainder that dom r ( G ) (cid:54) c dvrk r k .Define c lily := c lily , r,r, in the following. Claim.
Total r -Domination has a linear constraint core in BE classes. Proof.
Let L ⊆ V ( G ) be constraint core of G with | L | (cid:62) c lily dom r ( G ) . ByCorollary 1, we can find in polynomial time a uniform water lily ( R, C ) , C ⊆ L , | R | (cid:54) c lily of depth r , radius r , adhesion c and ratio . Let a ∈ C be anarbitrary centre, we claim that L \ { a } is still a constraint core, that is, everyset that totally r -dominates L \ { a } will also totally r -dominate a .To that end, let D be a minimal total r -dominating set and define D (cid:48) := D \ N rG − R [ C ] . If D (cid:48) totally r -dominates any part of C , it dominates all of C (andtherefore a ) as ( R, C ) is uniform. Similarly, if there exists u ∈ D ∩ N rG − R [ C ] suchthat u dominates at least two centres, then by uniformity it already dominatesall of C and in particular a . In all other cases, every set N rG − R [ a (cid:48) ] , a (cid:48) ∈ C mustcontain at least one vertex from D and we conclude that | D \ D (cid:48) | (cid:62) | C | (cid:62) | R | .Let ˜ D consist of D (cid:48) , R and up to | R | arbitrary neighbours R (cid:48) of R . We claimthat ˜ D is a total r -dominating set of G .Note that the only vertices that are not r -dominated by D (cid:48) lie inside N rG − R [ C ] —but this is precisely the set that is r -dominated by R . The vertices in R are dom-inated by R (cid:48) and vice-versa, we conclude that ˜ D is indeed totally r -dominating.We arrive at a contradiction since | D | = | D \ D (cid:48) | + | D (cid:48) | (cid:62) | R | + | D (cid:48) | > | R | + | D (cid:48) | (cid:62) | ˜ D | D to be minimal.Thus L \ { a } is a constraint core for Total r -Domination in G . We caniterate this procedure until | L | < c lily dom cr ( G ) and therefore end up with alinear constraint core.In the following, let L ⊆ V ( G ) be a constraint core for ( G, k ) with | L | (cid:54) c lily dom r ( G ) (cid:54) c lily c dvrk r k .We apply Lemma 4 with X = L , c = 1 , and r as here to obtain a projectionkernel ˆ G with | ˆ G | (cid:54) c total r,c | L | = O ( k ) . The proof that ( G, k ) is equivalent to theannotated instance ( ˆ G, L, k ) is almost identical to the proof in Theorem 5 andwe omit it here. Input:
A graph G , a set L ⊆ V ( G ) and an integer k . Problem:
Is there sets D , D ⊆ V ( G ) with | D | + 2 | D | (cid:54) k such that D r -dominates all of L \ D ? Annotated r -Roman Domination parametrised by k Theorem 7. r -Roman Domination over a hereditary BE-class G admits alinear bikernel into Annotated r -Roman Domination over the same class G .Moreover, the resulting graph is a ( r, -projection kernel of the original graph. Proof.
Let ( G, k ) be an input to r -Roman Domination where G is takenfrom a BE class. As a first step, we verify that dom r ( G ) is not too large bycomputing an r -dominating set using the algorithm described in Theorem 4. If thealgorithm returns a solution larger than c dvrk r k , we conclude that dom r ( G ) > k ,since ( G, k ) cannot be r -dominated by k vertices it in particular cannot be r -Roman dominated with that budget. In this case we output a trivial no-instance,thus assume for the remainder that dom r ( G ) (cid:54) c dvrk r k . Define c lily := c lily , r,r, in the following. Claim. r -Roman Domination has a linear constraint core in BE classes. Proof.
Let L ⊆ V ( G ) be constraint core of G with | L | (cid:62) c lily dom r ( G ) . ByCorollary 1, we can find in polynomial time a uniform water lily ( R, C ) , C ⊆ L , | R | (cid:54) c lily of depth r , radius r , adhesion c and ratio . Let a ∈ C be anarbitrary centre, we claim that L \ { a } is still a constraint core, that is, everyset that r -Roman-dominates L \ { a } will also r -Roman-dominate a .To that end, let D , D be a r -Roman dominating set of minimal cost( | D | + 2 | D | ). If | SP rR ( a ) ∩ D | (cid:62) this set already dominates a and there isnothing to prove, so assume otherwise. Then every centre a (cid:48) ∈ C must either becontained in D or have at least one D -vertex in its pad, e . g . | N rG − R [ a (cid:48) ] ∩ D | (cid:62) . Therefore the total cost of D , D when restricted to N rG − R [ C ] is at least | C | (cid:62) | R | .Construct the set D (cid:48) from D by removing all vertices in N rG − R [ C ] andconstruct the set D (cid:48) from D by removing all vertices in N rG − R [ C ] and adding15ll of R . Since R r -dominates all of N rG − R [ C ] and all these vertices are in D (cid:48) , wecan conclude that D (cid:48) , D (cid:48) is indeed an r -Roman dominating set. By our aboveobservation, the cost of D (cid:48) , D (cid:48) is at least | R | smaller than the cost of D , D ,contradiction minimality.Thus L \ { a } is a constraint core for ( r, c ) -Dominating Set in G . We caniterate this procedure until | L | < c lily dom r ( G ) and therefore end up with alinear constraint core.In the following, let L ⊆ V ( G ) be a constraint core for ( G, k ) with | L | (cid:54) c lily dom r ( G ) (cid:54) c lily c dvrk r k .We apply Lemma 4 with X = L , c = 1 , and r as here to obtain a projectionkernel ˆ G with | ˆ G | (cid:54) c total r,c | L | = O ( k ) . The proof that ( G, k ) is equivalent to theannotated instance ( ˆ G, L, k ) is almost identical to the proof in Theorem 5 andwe omit it here. Input:
A graph G , a set U ⊆ V ( G ) and an integer k . Problem:
Is there a set I ⊆ U of size at least k such that | N r [ v ] ∩ I | (cid:54) c for all v ∈ V ( G ) ? Annotated ( r, c ) -Scattered Set parametrised by kThe following proof makes use of the pad equivalence ∼ ν defined in Section 3:recall two centres u, v of a water lily ( R, C ) satisfy u ∼ ν v if they have the sameprojection-types onto R at the same distance (for distances smaller than thelily’s depth) inside their respective pads. Theorem 8. ( r, c ) -Scattered Set over a hereditary BE-class G admits alinear bikernel into Annotated ( r, c ) -Scattered Set over the same class G .Moreover, the resulting graph is an ( r, c ) -projection kernel of the original graph. Proof.
Let ( G, k ) be an instance of ( r, c ) -Scattered Set where G is takenfrom a BE class. As a first step, we deal with the case that sct cr ( G ) is large. Wecompute an c dvrk r -approximate r -dominating set D using Theorem 3. If | D | > c dvrk r wcol r ( G ) · k , we conclude by Theorems 1 and 2 that sct cr ( G ) (cid:62) sct r ( G ) > k and we output a trivial yes-instance. Otherwise, assume | D | (cid:54) c dvrk r wcol r ( G ) · k and define c lily := c lily , r,r, ,ν . We first show that ( r, c ) -Scattered Set admits alinear solution core. Claim. ( r, c ) -Scattered Set has a linear solution core in BE classes. Proof.
Let U ⊆ V ( G ) be solution core of G with | U | (cid:62) c lily dom r ( G ) . UsingCorollary 1, we find in polynomial time a ν -uniform water lily ( R, C ) , C ⊆ U , | R | (cid:54) c lily of depth r , radius r , adhesion and ratio . Let a ∈ C be anarbitrary centre, we claim that U \ { a } is still a solution core, i . e . there existsan optimal ( r, c ) -scattered set that does not contain a .To that end, let I be a minimum ( r, c ) -scattered set and assume a ∈ I . Weclaim that there exists an ( r, c ) -scattered set I (cid:48) of the same size which excludes a .16irst observe that every vertex that lives in a pad N r [ a (cid:48) ] , a (cid:48) ∈ C , has at least c neighbours in R at distance (cid:54) r . Therefore | N rG − R [ C ] ∩ I | (cid:54) | R | as otherwise wewould find a vertex in R whose r -neighbourhood contains more than c verticesof I . Since | C | (cid:62) | R | there are at least | R | centres C (cid:48) ⊆ C such that their pads N rG − R [ C (cid:48) ] do not intersect I . Since ( R, C ) is uniform and a ∈ I , we know that | N r [ a (cid:48) ] ∩ I | = | N r [ a ] ∩ I | < c for every centre a ∈ C .Take a (cid:48) ∈ C (cid:48) and let I (cid:48) := I \ { a } ∪ { a (cid:48) } . To see that I (cid:48) is ( r, c ) -scattered,consider any vertex u (cid:48) ∈ N r [ a (cid:48) ] (note that vertices at distance > r from a (cid:48) are not affected by the exchange of a by a (cid:48) ). By ν -uniformity, there existsa vertex u ∈ N r [ a ] with π rR [ u ] = π rR [ u (cid:48) ] . In particular, P rR ( u ) ∪ S rR ( u ) = P rR ( u (cid:48) ) ∪ S rR ( u (cid:48) ) ; therefore ( N r [ u ] ∩ I ) \ { a } = ( N r [ u (cid:48) ] ∩ I (cid:48) ) \ { a (cid:48) } and we concludethat | N r [ a (cid:48) ] ∩ I (cid:48) | (cid:54) c . It follows that U \ { a } is a solution core. We iterate theabove procedure until | U | (cid:54) c lily dom cr ( G ) and end up with a linear solutioncore.In the following, let U ⊆ V ( G ) be a solution core for ( G, k ) with | U | (cid:54) c lily dom r ( G ) (cid:54) c lily | D | = O ( k ) .We apply Lemma 4 with X = U and r , c as here to obtain a projectionkernel ˆ G with | ˆ G | (cid:54) c total r,c | U | = O ( k ) that a) preserves (cid:54) r -neighbourhoods in U and b) realizes every r -projection onto U that is realized p times in G at least min { c, p } times. Since distances in ˆ G [ U ] are as in G [ U ] , it is easy to see that anyset I ⊆ U is ( r, c ) -scattered in ˆ G iff it is ( r, c ) -scattered in G . Since U is furthera solution core for G , we conclude that ( G, k ) is equivalent to the annotatedinstance ( ˆ G, U, k ) .We show that ( r, [ λ, µ ]) -Domination admits a linear bikernel into the followingannotated problem: Input:
A graph G , sets L, U ⊆ V ( G ) and an integer k . Problem:
Is there a set D ⊆ U of size at most k such that | N r [ v ] ∩ D | (cid:62) λ for all v ∈ L and | N r [ v ] ∩ D | (cid:54) µ for all v ∈ V ( G ) ? Annotated ( r, [ λ, µ ]) -Domination parametrised by kWe note that the construction in the following proof results in a bikernel ( ˆ G, L, U, k ) with L ⊆ U , the construction can also be easily be modified toensure that L = U . Theorem 9. ( r, [ λ, µ ]) -Domination over a hereditary BE-class G admits alinear bikernel into Annotated ( r, [ λ, µ ]) -Domination over the same class G .Moreover, the resulting graph is an ( r, c ) -projection kernel of the original graph. Proof.
Since the cases where either µ = ∞ or λ = 0 are equivalent to ( r, c ) -Dominating Set or ( r, c ) -Scattered Set and thus covered by Theorems 5and 8, we here only consider the case of λ (cid:54) = 0 and µ (cid:54) = ∞ .Note that any solution to the problem is in particular an ( r, µ ) -dominatingset. As a first step, we therefore deal with the case that dom µr ( G ) is toolarge by computing an ( r, µ ) -dominating set using the algorithm described in17heorem 4. If the algorithm returns a solution larger than c cdom r,c k , we concludethat dom µr ( G ) > k and therefore that ( G, k ) must be a no-instance; in whichcase we output a trivial no-instance. Otherwise, let ˆ D be the resulting ( r, c ) -dominating set.Let ( G, L, U, k ) be an instance of Annotated ( r, [ λ, µ ]) -Domination with L = U = V ( G ) . Clearly, ( G, L, U, k ) is equivalent to ( G, k ) . In the following, wegradually reduce the size of L and U while maintaining this equivalence. Tothat end, we will use the pad signature ν which is to be understood to take the‘vertex labels’ L , U into account.Assume that | L | > ( c lily + 1) | ˆ D | with c lily := c lily r, r,µ +1 ,ν . Then, using ˆ D in theconstruction used in the proof of Lemma 6, we find a ν -uniform water lily ( R, C ) with C ⊆ L \ ˆ D of depth r , radius r and ratio ( µ + 1) . Claim.
Let a ∈ C . Then the instances ( G, L, U, k ) and ( G, L \ { a } , U, k ) areequivalent. Proof.
Any solution for ( G, L, U, k ) is also a solution to ( G, L \ { a (cid:48) } , U, k ) , there-fore we only have to show the opposite direction.Let D be a solution for ( G, L \ { a } , U, k ) . Since R ⊆ L ∩ U , the set D canintersect at most µ | R | pads or otherwise we would violate an upper constraint forat least one of the vertices in R . It follows that at least | R | pads of ( R, C ) cannotcontain any vertex of D ; let the centres of these pads be C (cid:48) ⊆ C . Choose a (cid:48) ∈ C (cid:48) distinct from a (since | C (cid:48) | (cid:62) | R | (cid:62) λ > such a vertex exists). Note that a (cid:48) ∈ L , therefore | N r [ a (cid:48) ] ∩ D | (cid:62) λ . But since N rG − R [ a (cid:48) ] ∩ D = ∅ , these solutionvertices must lie in SP rR ( a (cid:48) ) . Now simply observe that, by uniformity of ( R, C ) , SP rR ( a ) = SP rR ( a (cid:48) ) and therefore | N r [ a (cid:48) ] ∩ D | (cid:62) | SP rR ( a ) ∩ D | (cid:62) λ . Accordingly, D is also a solution for ( G, L, U, k ) .We repeat the above procedure until | L \ ˆ D | (cid:54) c lily k . Now assume that | U \ ( L ∪ ˆ D ) | > c lily k and let ( R, C ) be a ν -uniform water lily with C ⊆ U \ ( L ∪ ˆ D ) ofdepth r , radius r and ratio ( µ + 1) | R | . Claim.
Let a ∈ C . Then the instances ( G, L, U, k ) and ( G, L, U \ { a } , k ) areequivalent. Proof.
By construction of ( R, C ) , every vertex x ∈ N rG − R [ C ] is ( r, µ ) -dominatedby R ∩ ˆ D . Importantly, R ∩ ˆ D ⊆ R ∩ U , therefore any solution D of ( G, L, U, k ) can intersect N r [ R ] in at most µ | R | vertices. In particular, at most µ | R | padsof ( R, C ) can contain vertices of D , let us call the centres of these emptypads C (cid:48) ⊆ C .If a (cid:54)∈ D , clearly D is a solution of ( G, L, U \ { a } , k ) and there is nothing toprove. Assume therefore that a ∈ D . Let a (cid:48) ∈ C (cid:48) be an arbitrary centre of anempty pad. We claim that D (cid:48) := D \ { a } ∪ { a (cid:48) } is a solution to ( G, L, U \ { a } , k ) .To that end, consider any vertex x ∈ N r [ a ] ∪ N r [ a (cid:48) ] , we will show that D (cid:48) fulfilsany constraints associated with x . Case 1 . x ∈ N rG − R [ a ] .By ν -uniformity, there exists a vertex x (cid:48) ∈ N rG − R [ a (cid:48) ] such that SP rG − R ( x ) = P rG − R ( x (cid:48) ) and x (cid:48) is contained in L ( U ) iff x is contained in L ( U ). For thespecial case that x = a we let x (cid:48) = a (cid:48) .Assume x ∈ L , then x (cid:48) ∈ L and accordingly | N r [ x (cid:48) ] ∩ D | (cid:62) λ . Since N rG − R [ a (cid:48) ] ∩ D = ∅ , we have that N r [ x (cid:48) ] ∩ D = SP rR ( x (cid:48) ) ∩ D = SP rR ( x (cid:48) ) ∩ D (cid:48) = SP rR ( x ) ∩ D (cid:48) , therefore | N r [ x ] ∩ D (cid:48) | = | N r [ x (cid:48) ] ∩ D | (cid:62) λ and the lower-bound constraint for x is satisfied by D (cid:48) .If x ∈ R , simply note that | N r [ x ] ∩ D (cid:48) | (cid:54) | N r [ x ] ∩ D | (cid:54) µ , hence theupper-bound constraint for x is satisfied by D (cid:48) . Case 2 . x ∈ N rG − R [ a (cid:48) ] Again, by ν -uniformity, there exists a vertex ˆ x ∈ N rG − R [ a ] such that SP rG − R ( x ) = SP rG − R (ˆ x ) and ˆ x is contained in L ( U ) iff x iscontained in L ( U ). For the special case that x = a (cid:48) we let ˆ x = a .If x ∈ L , simply note that | N r [ x ] ∩ D (cid:48) | (cid:62) | N r [ x ] ∩ D | (cid:62) λ , hence thelower-bound constraint for x is satisfied by D (cid:48) .Assume x ∈ R . Then ˆ x ∈ R and accordingly | N r [ˆ x ] ∩ D | (cid:54) µ . Morespecifically, since a ∈ N r [ˆ x ] ∩ D , we know that | SP rR [ˆ x ] ∩ D | (cid:54) µ − . Because N r [ x ] ∩ D (cid:48) = ( SP rR [ x ] ∩ D (cid:48) ) ∪ { a (cid:48) } = ( SP rR [ˆ x ] ∩ D ) ∪ { a (cid:48) } we conclude that | N r [ x ] ∩ D (cid:48) | (cid:54) µ and the upper-bound constraint for x issatisfied by D (cid:48) . Case 3 . x ∈ SP rR ( a ) = SP rR ( C ) . Simply note that by uniformity | N r [ x ] ∩ D | = | N r [ x ] ∩ D (cid:48) | and therefore D (cid:48) satisfies all constraints for x .Therefore D (cid:48) is indeed a solution for ( G, L, U \ { a } , k ) of equal size and weconclude that the instances ( G, L, U, k ) and ( G, L, U \ { a } , k ) are equivalent, asclaimed.We repeat the above procedure until | U \ ( L ∪ ˆ D ) | (cid:54) c lily k and end up with aninstance ( G, L, U, k ) which is equivalent to our initial instance ( G, k ) and furthersatisfies | L | (cid:54) c lily k and | U | (cid:54) | L | + | ˆ D | + | U \ ( L ∪ ˆ D ) | (cid:54) (2 c lily + c cdom r,c ) k .Finally, let us construct the bikernel from this annotated instance. Notethat, by construction, L ⊆ U . Let ˆ U be the shortest-path closure of U in G asper Lemma 3, then | ˆ U | (cid:54) c pathcl r | U | and ˆ G := G [ ˆ U ] preserves all distances upto length r between vertices in U . In particular, N r ˆ G [ v ] ∩ U = N rG [ v ] ∩ U . Sincethe annotated instance asks for solutions contained entirely in U and L ⊆ U , weconclude that the instance ( G, L, U, k ) and ( ˆ G, L, U, k ) are equivalent, thereforethe latter is also equivalent to ( G, k ) which finally proves the claim. If we sacrifice the constraint to construct a (bi)kernel that is contained in thesame hereditary graph class, we are able to construct BE-kernels by reducingfrom the annotated problem back into the original problems. In the following19onstructions, we usually tried to minimize the increase in the parameter k , notthe increase of the expansion characteristics of the class. Theorem 10. ( r, c ) -Dominating Set admits a linear BE-kernel. Proof.
For an instance ( G, k ) of ( r, c ) -Dominating Set , where G is takenfrom a BE class, we first construct a bikernel ( ˆ G, L, k ) of Annotated ( r, c ) -Dominating Set according to Theorem 5. Recall that ˆ G is an ( r, c ) -projectionkernel of ( G, L ) .First consider r (cid:62) . We construct G (cid:48) from ˆ G by adding new vertices a , . . . , a c , b , b , b to the graph. We connect every a i , (cid:54) i (cid:54) c to both b and b ; then connect b to every vertex in O := V ( ˆ G ) \ L via a path of length r − and connect b to b by such a path as well.From the construction it is clear that G (cid:48) has size O ( k ) , we are left withproving that the two instances ( G, k ) and ( G (cid:48) , k + c ) are equivalent.Assume that D (cid:48) is a minimum ( r, c ) -dominating set for G (cid:48) of size (cid:54) k + c .By a simple exchange argument, we can assume that D (cid:48) contains all vertices a i in order to ( r, c ) -dominate b . These vertices already ( r, c ) -dominate all of O and the paths leading from b to O . As such, we can assume that an optimalsolution D (cid:48) does not contain internal vertices of those paths (otherwise we mightas well exchange an internal vertex for the path’s endpoint in O ). Then theset ˆ D := D (cid:48) \ { a , . . . a c } has size at most k and ( r, c ) -dominates all of L ; thus ˆ D in particular is a solution to ( ˆ G, L, k ) .In the other direction, assume that ˆ D is a minimum solution for ( ˆ G, L, k ) ,that is, ˆ D ( r, c ) -dominates L in ˆ G . Let D (cid:48) := ˆ D ∪ { a , . . . , a c } , it is easy to seethat D (cid:48) ( r, c ) -dominates G (cid:48) and has size | D (cid:48) | = | D | + c .For r = 1 we modify the construction as follows: we add vertices a , . . . , a c , b and connect all a i to O ∪ { b } . The argument for why the resulting instance isequivalent is very similar to the case r (cid:62) and we omit it here.We conclude that ( ˆ G, L, k ) and ( G (cid:48) , k + c ) are indeed equivalent, and thusalso to ( G, k ) . It is only left to show that the construction of G (cid:48) increasedthe expansion characteristics by some arbitrary function independent of | G | .Simply note that we can construct G (cid:48) from G by adding c + 3 apex-vertices(which increases the expansion characteristics only by an additive constant) andthen remove or subdivide edges incident to them (which does not increase theexpansion characteristics). Theorem 11.
Total r -Domination admits a linear BE-kernel. Proof.
For an instance ( G, k ) of Total r -Domination we first construct abikernel ( ˆ G, L, k ) of Annotated Total r -Domination according to Theorem 6.Recall that ˆ G is an ( r, -projection kernel of ( G, L ) .We construct G (cid:48) from ˆ G as follows: add new vertices b, a , a to the graph.Connect b to every vertex in O := V ( ˆ G ) \ L and to a via a path of length r . Then connect a to a by a path of length r . It is is clear that G (cid:48) hassize O ( k ) , we are left with proving that the two instances ( G, k ) and ( G (cid:48) , k + 2) are equivalent. 20rom the construction it is clear that G (cid:48) has size O ( k ) , we are left withproving that the two instances ( G, k ) and ( G (cid:48) , k + 2) are equivalent.First, assume that D (cid:48) is a minimal total r -dominating set for G (cid:48) . Since thepath from b to a needs to contain at least one vertex to dominate the path, wecan, by a simple exchange argument, assume that this vertex is a . D (cid:48) furtherneeds to dominate a itself, again by an exchange argument we may assumethat b ∈ D (cid:48) . We can therefore assume that D (cid:48) does not contain the pathsbetween b and O (excluding the vertices O ) and the path from b to a in verticesother than b, a . Then the set ˆ D := D (cid:48) \ { b, a } has size | D (cid:48) | − and totally r -dominates all of L , therefore ˆ D is a solution to ( ˆ G, L, k ) .In the other direction, assume that ˆ D is a minimal solution for ( ˆ G, L, k ) ,that is, ˆ D totally r -dominates L in ˆ G . Let D (cid:48) := ˆ D ∪ { b, a } . Then D (cid:48) totally r -dominates G (cid:48) and has size | D | + 2 .We conclude that ( ˆ G, L, k ) and ( G (cid:48) , k + 2) are indeed equivalent, and thelatter is also equivalent to ( G, k ) . The argument the increase of the expansioncharacteristic of G (cid:48) is similar to before, we omit it here. Theorem 12. r -Roman Domination admits a linear BE-kernel. Proof.
For an instance ( G, k ) of r -Roman Domination we first construct a bik-ernel ( ˆ G, L, k ) of Annotated r -Roman Domination according to Theorem 7.Recall that ˆ G is an ( r, -projection kernel of ( G, L ) .We construct G (cid:48) from ˆ G as follows: add new vertices b, a , a , a to the graph.Connect b to every vertex in O := V ( ˆ G ) \ L and to { a , a , a } via a path oflength r . It is is clear that G (cid:48) has size O ( k ) , we are left with proving that thetwo instances ( G, k ) and ( G (cid:48) , k + 2) are equivalent.First, assume that D (cid:48) , D (cid:48) is a minimal r -Roman dominating set for G (cid:48) ofsize at most k + 2 . By a simple exchange argument, we can assume that b ∈ D (cid:48) in order to r -Roman-dominate a , a , and a (including all three vertices in D (cid:48) would be more expensive). Now b already r -Roman-dominates all of O as wellas the paths added during the construction, we can therefore assume that D (cid:48) isentirely contained in V ( G ) . Therefore the sets D (cid:48) , D (cid:48) \ { b } r -Roman-dominates L in G at a cost of | D (cid:48) | + 2 | D (cid:48) | − .In the other direction, assume that ˆ D , ˆ D is a minimal-cost solution for ( ˆ G, L, k ) ,that is, ˆ D , ˆ D r -Roman-dominates L in ˆ G . Partition both set ˆ D i for i ∈ [1 , into sets ˆ D i,O = ˆ D i \ L and ˆ D i,L = ˆ D i ∩ L . Then construct D (cid:48) i,O as follows:for every equivalence class [ u ] ∈ ˆ D i,O / ∼ rL , include a vertex of [ u ] ∩ O in D (cid:48) i,O (here we use that ˆ G is am ( r, -projection kernel of ( G, L ) . Since we picked thesame projection-classes as in ˆ D ,O , ˆ D ,O , we conclude that the sets D (cid:48) ,O ∪ ˆ D ,L , D (cid:48) ,O ∪ ˆ D ,L r -Roman-dominate the core L . Therefore, the sets D (cid:48) := D (cid:48) ,O ∪ ˆ D ,L , D (cid:48) := D (cid:48) ,O ∪ ˆ D ,L ∪ { b } r -Roman-dominate all of G (cid:48) at cost of | ˆ D | + 2 | ˆ D | + 2 .We conclude that ( ˆ G, L, k ) and ( G (cid:48) , k + 2) are indeed equivalent, and thelatter is also equivalent to ( G, k ) . To see that the expansion characteristics21nly increase by a function that is independent of | G | , simply note that we canconstruct G (cid:48) by adding one apex-vertex to G with an additional pendant vertex(which increases the expansion characteristics only by an additive constant)and then subdivide edges incident to it (which does not increase the expansioncharacteristics). Theorem 13. ( r, c ) -Scattered Set admits a linear BE-kernel. Proof.
Let ( G, k ) be an input of ( r, c ) -Scattered Set where G is taken froma BE class. We first construct the annotated bikernel ( ˆ G, U, k ) according toTheorem 8 and then construct G (cid:48) from ˆ G by adding vertices a , a , b , . . . , b c and edges a b i for all (cid:54) i (cid:54) c . We further connect a to all vertices in O := V ( ˆ G ) \ U via paths of length r and to a via a path of length r − (for r = 1 we identify a and a ). It is is clear that G (cid:48) has size O ( k ) , we are left toprove that the instances ( ˆ G, U, k ) and ( G (cid:48) , k + c ) are equivalent.First, consider a maximal ( r, c ) -scattered set I (cid:48) in G (cid:48) . Since O ∪{ b , . . . , b c } ⊂ N r [ a ] we may assume, by a simple exchange argument, that { b , . . . , b c } ⊆ I (cid:48) .Accordingly, O ∩ I (cid:48) = ∅ and I := I (cid:48) \{ b , . . . , b c } is an ( r, c ) -scattered set containedentirely in U . Therefore I is ( r, c ) -scattered in ˆ G as well and | I | = | I (cid:48) | + c .In the other direction, assume that ˆ I ⊆ U is a maximal ( r, c ) -scattered set in ˆ G .Then N rG (cid:48) [ a ] ∩ I = ∅ and we can add up to c vertices from N r [ a ] to I . Since thevertices b i all lie at distance r from O , we conclude that I (cid:48) := I ∪ { b , . . . , b c } is indeed ( r, c ) -scattered in G (cid:48) and | I (cid:48) | = | I | + c .We conclude that the instances ( ˆ G, U, k ) and ( G (cid:48) , k + c ) are equivalent andhence ( G, k ) and ( G (cid:48) , k + c ) are as well. The argument why the expansioncharacteristics only increase by a constant are similar to the arguments inTheorem 10. Theorem 14. r -Perfect Code admits a linear BE-kernel. Proof.
Let ( G, k ) be an input instance of r -Perfect Code where G is takenfrom a BE class. Since r -Perfect Code is equivalent to ( r, [1 , -Domination ,we proceed by first constructing the annotated bikernel ( ˆ G, L, U, k ) accordingto Theorem 9. As commented there, we can construct the bikernel that L = U which we will assume in the following for simplicity.Let O := V ( ˆ G ) \ L . We construct G (cid:48) from ˆ G by appending a path P u of length r to every vertex u ∈ O . We claim that ( ˆ G, L, U, k ) is equivalentto ( G (cid:48) , k + | O | ) . In the following, fix one path P u and let a , . . . , a r be itsvertices ordered by their respective distance from u ; the arguments we make willhold symmetrical for all paths added in the construction.First, consider an r -perfect code D (cid:48) of G (cid:48) . In order to dominate the vertex a r ,it needs to contain a vertex a j ∈ P u with r (cid:54) j (cid:54) r . Since a j will in particulardominate a r , we conclude that u (cid:54)∈ D (cid:48) and, by symmetry, that D (cid:48) ∩ O = ∅ . Thenthe set ˆ D := D (cid:48) ∩ V ( ˆ G ) is indeed a perfect code for ˆ G of size | D (cid:48) | − | O | .In the other direction, assume that ˆ D ⊆ L is a perfect code for L in ˆ G .Since L = U is both a solution- and a constraint core for G , we know that theset ˆ D is a perfect code in G . Because ˆ G is an induced subgraph of G , we conclude22hat | N r [ u ] ∩ ˆ D | (cid:54) for all u ∈ O . Let d u be the distance of u ∈ O to theclosest vertex in ˆ D (this distance is, by construction, the same in ˆ G and G (cid:48) ). Weconstruct D (cid:48) from ˆ D as follows: if d u > r , then we add the vertex a r . Otherwise,note that the vertices a , . . . , a i for i = r − d u of P u are dominated by ˆ D , wetherefore add the vertex a j with j = 2 r − d u + 1 . The resulting set D (cid:48) dominates,in G (cid:48) , all vertices in O that are not dominated by ˆ D and further dominates allvertices V ( G (cid:48) ) \ V ( ˆ G ) precisely once. It follows that D (cid:48) is a perfect code in G (cid:48) of size | ˆ D | + | O | .We conclude that the instances ( ˆ G, L, U, k ) and ( G (cid:48) , k + | O | ) are equivalentand hence ( G, k ) and ( G (cid:48) , k + | O | ) are as well. The construction of G (cid:48) from ˆ G only increases the expansion characteristics if the original graph class consist ofedgeless graphs. The following results are applicable to graph BE-classes that are closed underthe addition of pendant vertices, e . g . planar graphs, graphs of bounded genusor graph classes defined by an excluded minor of minimum degree two. Theirproofs are a collection of arguments already made in detail in Section 5, wewill abbreviate those parts here. In the following, let dom total r ( G ) denotes thetotal r -domination number and dom roman r ( G ) the r -Roman domination numberof G . We will also write dom r ( G, L ) , dom total r ( G, L ) , and dom roman r ( G, L ) forthe annotate domination numbers (where only the set L ⊆ V ( G ) has to bedominated). Theorem 15.
Let G be a hereditary BE-class that is further closed underadding pendant vertices. Given a graph G ∈ G and an integer r we can computein polynomial time a graph G (cid:48) ∈ G and an integer c with the following properties: • | G (cid:48) | = O ( dom r ( G )) = O ( dom totalr ( G )) = O ( dom romanr ( G )) , • dom r ( G (cid:48) ) = dom r ( G ) + c , • dom total r ( G (cid:48) ) = dom total r ( G ) + c , and • dom roman r ( G (cid:48) ) = dom roman r ( G ) + 2 c . Proof.
We apply the constructions from Theorems 5 (for c = 1 ), 6, and 7 tofind constraint cores L d , L t and L r for all three problems. Since dom total r and dom roman r lie within a factor of two of dom r , we conclude that the joint set K := L d ∪ L t ∪ L r is a constraint core for all three problems of size | K | = O ( dom r ( G )) .Let ˆ G be a ( µ, -projection kernel of ( G, K ) constructed according toLemma 4, recall that ˆ G is an induced subgraph of G with | ˆ G | = O ( | K | ) . Bythe proofs of Theorems 5, 6, and 7 we have that dom r ( ˆ G, L ) = dom r ( G ) , dom total r ( ˆ G, L ) = dom total r ( G ) , and dom roman r ( ˆ G, L ) = dom roman r ( G ) .Let O := V ( ˆ G ) \ K be the vertices outside the core set K and let c := 2 | O | .Let T be the tree constructed as follows: create vertices b , b , b , a , . . . , a .Connect b to b and b to b by paths of length r , then connect b to a , . . . , a r and b to a , . . . , a . We construct G (cid:48) by appendingto each vertex v ∈ O a copy T v of T by identifying b with v . It is not difficultto see that any optimal r -dominating set and total r -dominating set can, by anexchange argument, be assumed to contain the vertices b and b of each tree T v ;and that any r -Roman-dominating set includes b and b at a cost of twoeach. We conclude that dom r ( G (cid:48) ) = dom r ( ˆ G, L ) + c = dom r ( G ) + c, dom total r ( G (cid:48) ) = dom total r ( ˆ G, L ) + c = dom total r ( G ) + c, and dom roman r ( G (cid:48) ) = dom roman r ( ˆ G, L ) + 2 c = dom roman r ( G ) + 2 c. Recall that an r -scattered set is equivalent to a r -independent set and inparticular that sct r ( G ) = ind r ( G ) . Theorem 16.
Let G be a hereditary BE-class that is further closed underadding pendant vertices. Given a graph G ∈ G and integers λ (cid:54) µ we cancompute in polynomial time a graph G (cid:48) ∈ G and integers c λ , . . . , c µ with thefollowing properties: • | G (cid:48) | = O ( dom λ ( G )) , • for all λ (cid:54) r (cid:54) µ it holds that dom r ( G (cid:48) ) = dom r ( G ) + c r , and • for all λ (cid:54) r (cid:54) µ it holds that ind r ( G (cid:48) ) = ind r ( G ) + c r . Proof.
We apply the constructions from Theorems 5 and 8 for r ∈ [ λ, µ ] toconstruct constraint cores L r for r -Dominating Set and solution cores S r for r -Scattered Set . Let L := (cid:83) λ (cid:54) r (cid:54) µ L r and S := (cid:83) λ (cid:54) r (cid:54) µ S r ; since | L r | = O ( dom r ( G )) and | S r | = O ( ind r ( G )) and ind r ( G ) = Θ( ds r ( G )) by Theorem 1we conclude that | L ∪ S | = O (( µ − λ ) dom λ ( G )) = O ( dom λ ( G )) . Define K := L ∪ S and note that K is a constraint core for r -Dominating Set and asolution core for r -Scattered Set for all λ (cid:54) r (cid:54) µ .Let ˆ G be a ( µ, -projection kernel of ( G, K ) constructed according toLemma 4, recall that ˆ G is an induced subgraph of G with | ˆ G | = O ( | K | ) .Let O := V ( ˆ G ) \ K be the vertices outside the core set K . By construction,note that any minimal r -dominator of K in ˆ G has size dom r ( G ) and that anymaximal r -scattered set of ˆ G contained in K as size sct r ( G ) for all λ (cid:54) r (cid:54) µ .Let σ be an integer divisible by all integers (2 r + 1) for λ (cid:54) r (cid:54) µ . Weconstruct G (cid:48) from ˆ G by appending a path of length σ − to every vertex v ∈ O and call the resulting path (including v ) P v . The size of G (cid:48) is boundedby O ( | K | ) = O ( dom λ ( G )) , it remains to show the second property.Fix r ∈ [ µ, λ ] and define c r := σ r +1 | O | . First assume that D is a minimal r -dominating set of G . Then, as in the proof of Theorem 5, there exists aset ˆ D of the same size that r -dominates K in ˆ G . We construct D (cid:48) from ˆ D byincluding σ/ (2 r + 1) vertices of each path P v ; namely all vertices at position i (2 r + 1) − r , (cid:54) i (cid:54) σ/ (2 r + 1) in P v (where v has position 1). Since these24ertices dominate all of P v , we conclude that D (cid:48) dominates all of G (cid:48) and hassize | D (cid:48) | = | ˆ D | + σ r +1 | O | = | D | + c r .In the other direction, let D (cid:48) be a minimum r -dominating set for G (cid:48) . Collectthe vertices of D (cid:48) that lie on P v \ { v } in the set D (cid:48) P . By a simple exchangeargument D (cid:48) P intersects every path P v in the same indices as above, i . e . thevertices at positions i (2 r + 1) − r , (cid:54) i (cid:54) σ/ (2 r + 1) . It follows that | D (cid:48) P | = c r .Note that D (cid:48) P cannot r -dominate any vertex in K , hence D := D (cid:48) \ D (cid:48) P must r -dominate all of K and by construction of G (cid:48) this also holds true in thegraph ˆ G . Since K is a constraint core for G , we conclude that D r -dominatesall of G and has size | D | = | D (cid:48) | − | D (cid:48) P | = | D (cid:48) | − c r . We conclude that indeed dom r ( G (cid:48) ) = dom r ( G ) + c r .Now consider a maximal r -scattered set I of G . Then, as in the proof ofTheorem 8, there exists a set ˆ I ⊆ K which is r -scattered in ˆ G . We construct I (cid:48) from ˆ I by including σ/ (2 r + 1) vertices of each path P v ; namely all verticesat position i (2 r + 1) , (cid:54) i (cid:54) σ/ (2 r + 1) in P v . Since ˆ I is disjoint with O , theresulting set is indeed r -scattered and has size | ˆ I | := | I | + σ r +1 | O | = | I | + c r .In the other direction, let I (cid:48) be a maximal r -scattered set in G (cid:48) . By asimple exchange argument, we can assume that I (cid:48) contains all endpoints ofthe paths P v , v ∈ O and, by repeating this argument, we can assume that I (cid:48) intersects every path P v at precisely the positions i (2 r + 1) , (cid:54) i (cid:54) σ/ (2 r + 1) .Collect this part of I (cid:48) in the set I (cid:48) P , note that | I (cid:48) P | = c r . We further conclude that O ∩ I (cid:48) = ∅ , therefore the set I := I (cid:48) \ I (cid:48) P is completely contained in K and I is r -scattered in ˆ G . Since K is a solution core, it follows that K is also r -scatteredin G and we have that | I | = | I (cid:48) | − | I (cid:48) P | = | I (cid:48) | − c r . We conclude that indeed sct r ( G (cid:48) ) = sct r ( G ) + c r and therefore ind r ( G (cid:48) ) = ind r ( G ) + c r . We defined the notion of water lilies and showed that in BE-classes thesestructures can be used to compute linear-sized cores, bikernels, and BE-kernels.These constructions are almost universal, to the point were we can combinethem into ‘multikernels’. It stands to reason that there might be a generalformulation for these types of kernels. 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