Cycles
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Akanksha Agrawal
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Institute of Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI),
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Budapest, Hungary
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akanksha@sztaki.mta.hu
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Pallavi Jain
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Institute of Mathematical Sciences, HBNI, Chennai, India
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pallavij@imsc.res.in
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Lawqueen Kanesh
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Institute of Mathematical Sciences, HBNI, Chennai, India
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lawqueen@imsc.res.in
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Pranabendu Misra
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University of Bergen, Bergen, Norway
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Pranabendu.Misra@uib.no
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Saket Saurabh
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Institute of Mathematical Sciences, HBNI, Chennai, India
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saket@imsc.res.in
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Abstract
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A generalization of classical cycle hitting problems, called conflict version of the problem, is
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defined as follows. An input is undirected graphsGandHon the same vertex set, and a positive
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integer k, and the objective is to decide whether there exists a vertex subsetX ⊆ V(G) such
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that it intersects all desired “cycles” (all cycles or all odd cycles or all even cycles) andX is an
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independent set inH. In this paper we study the conflict version of classicalFeedback Vertex
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Set, andOdd Cycle Transversalproblems, from the view point of kernelization complexity.
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In particular, we obtain the following results, when the conflict graphH belongs to the family
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ofd-degenerate graphs.
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1. CF-FVSadmits aO(kO(d)) kernel.
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2. CF-OCTdoes not admit polynomial kernel (even whenHis 1-degenerate), unlessNP⊆ coNPpoly.
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For our kernelization algorithm we exploit ideas developed for designing polynomial kernels for
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the classicalFeedback Vertex Setproblem, as well as, devise new reduction rules that exploit
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degeneracy crucially. Our main conceptual contribution here is the notion of “k-independence
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preserver”. Informally, it is a set of “important” vertices for a given subset X ⊆ V(H), that
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is enough to capture the independent set property in H. We show that ford-degenerate graph
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independence preserver of sizekO(d) exists, and can be used in designing polynomial kernel.
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2012 ACM Subject Classification Theory of computation→Design and analysis of algorithms
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→Parameterized complexity and exact algorithms
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Keywords and phrases Parameterized Complexity, Kernelization, Conflict-free problems, Feed-
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back Vertex Set, Even Cycle Transversal, Odd Cycle Transversal
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Digital Object Identifier 10.4230/LIPIcs.IPEC.2018.14
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Funding This research has received funding from the European Research Council under ERC
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grant no. 306992 PARAPPROX, ERC grant no. 715744 PaPaALG and ERC grant no. 725978
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SYSTEMATIC-GRAPH, and DST, India for SERB-NPDF fellowship [PDF/2016/003508].
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© A. Agrawal and P. Jain and L. Kanesh and P. Misra and S. Saurabh;
1 Introduction
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Reducing the input data, in polynomial time, without altering the answer is one of the
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popular ways in dealing with intractable problems in practice. While such polynomial
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time heuristics can not solve NP-hard problems exactly, they work well on input instances
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arising in real-life. It is a challenging task to assess the effectiveness of such heuristics
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theoretically. Parameterized complexity, via kernelization, provides a natural way to quantify
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the performance of such algorithms. In parameterized complexity each problem instance
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comes with a parameterk and the parameterized problem is said to admit a polynomial
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kernelif there is a polynomial time algorithm, called a kernelizationalgorithm, that reduces
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the input instance down to an instance with size bounded by a polynomialp(k) ink, while
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preserving the answer. The reduced instance is called ap(k) kernel for the problem.
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The quest for designing polynomial kernels for “hitting cycles” in undirected graphs has
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played significant role in advancing the field of polynomial time pre-processing – kernelization.
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Hitting all cycles, odd cycles and even cycles correspond to well studied problems ofFeedback
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Vertex Set(FVS),Odd Cycle Transversal(OCT) and Even Cycle Transversal
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(ECT), respectively. Alternatively,FVS,OCTandECTcorrespond to deleting vertices such
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that the resulting graph is a forest, a bipartite graph and an odd cactus graph, respectively.
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All these problems,FVS,OCT, andECT, have been extensively studied in parameterized
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algorithms and kernelization. The earliest knownFPTalgorithms forFVS go back to the
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late 80’s and the early 90’s [4, 11] and used the seminal Graph Minor Theory of Robertson
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and Seymour. On the other hand the parameterized complexity ofOCTwas open for long
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time. Only, in 2003, Reed et al. [24] gave a 3knO(1) time algorithm forOCT. This is also
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the paper which introduced themethod of iterative compressionto the field of parameterized
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complexity. However, the existence of polynomial kernel, for FVSandOCT were open
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questions for long time. ForFVS, Burrage et al. [7] resolved the question in the affirmative
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by designing a kernel of sizeO(k11). Later, Bodlaender [5] reduced the kernel size toO(k3),
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and finally Thomassé [25] designed a kernel of sizeO(k2). The kernel of Thomassé [25] is
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best possible under a well known complexity theory hypothesis. It is important to emphasize
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that [25] popularized the method ofexpansion lemma, one of the most prominent approach
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in designing polynomial kernels. While, the kernelization complexity ofFVSwas settled
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in 2006, it took another 6 years and a completely new methodology to design polynomial
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kernel forOCT. Kratsch and Wahlström [16] resolved the question of existence of polynomial
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kernel forOCTby designing a randomized kernel of sizeO(k4.5) using matroid theory.1 As
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a counterpart toOCT, Misra et al. [20] studiedECTand designed anO(k3) kernel.
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Fruitful and productive research on FVS andOCT have led to the study of several
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variants and generalizations of FVS andOCT. Some of these admit polynomial kernels
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and for some one can show that none can exist, unless some unlikely collapse happens in
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complexity theory. In this paper we study the following generalization ofFVS, andOCT,
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from the view-point of kernelization complexity.
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Conflict Free Feedback Vertex Set(CF-FVS) Parameter: k Input:An undirected graphG, a conflict graphHon vertex setV(G) and a non-negative integerk.
Question: Does there existS⊆V(G), such that|S| ≤k,G−S is a forest andH[S] is edgeless?
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1 This foundational paper has been awarded the Nerode Prize for 2018.
One can similarly define Conflict Free Odd Cycle Transversal(CF-OCT).
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Motivation. On the outset, a natural thought is “why does one care” about such an
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esoteric (or obscure) problem. We thoughtexactly the same in the beginning, till we realized
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the modeling power the problem provides and the rich set of questions one can ask. In the
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course of this paragraph we will try to explain this. First observe that, if one wants to model
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“independent” version of these problems (where the solution is suppose to be an independent
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set), then one takes conflict graph to be same as the input graph. An astute reader will figure
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out that the problem as stated above is W[1]-hard – a simple reduction fromMulticolor
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Independent Set with each color class being modeled as cycle and the conflict graph
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being the input graph. Thus, a natural question is: when does the problem become FPT? To
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state the question formally, letF andG be two families of graphs. Then, (G,F)-CF-FVSis
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same problem asCF-FVS, but the input graphGand the conflict graphH are restricted
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to belong toG and H, respectively. It immediately brings several questions: (a) for which
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pairs of families the problem isFPT; (b) can we obtain some kind of dichotomy results; and
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(c) what could we say about the kernelization complexity of the problem. We believe that
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answering these questions for basic problems such as FVS, OCT, andDominating Set
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will extend both the tractability as well as intractability tools in parameterized complexity
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and led to some fruitful and rewarding research. It is worth to note that initially we were
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inspired to define these problems by similar problems in computational geometry. See related
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results for more on this.
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Our Results and Methods. A graphGis calledd-degenerateif every subgraph of G
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has a vertex of degree at most d. For a fixed positive integerd, letDd denote the set of
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graphs ofdegeneracyat most d. In this paper we study the (?,Dd)-CF-FVS (Dd-CF-FVS)
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problem. The symbol ? denotes that the input graph G is arbitrary. One can similarly
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define Dd-CF-OCT. In fact, we study, CF-OCT for a very restricted family of conflict
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graphs, a family of disjoint union of paths of length at most three and at most two star
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graphs. We denote this family as P≤3?? and this variant of CF-OCT as P≤3??-CF-OCT.
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Starting point of our research is the recent study of Jain et al. [14], who studied conflict-free
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graph modification problems in the realm of parameterized complexity. As a part of their
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study they gaveFPTalgorithms forDd-CF-FVS,Dd-CF-OCTandDd-CF-ECTusing the
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independence covering families [17]. Their results also imply similarFPTalgorithm when the
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conflict graph belongs to nowhere dense graphs. In this paper we focus on the kernelization
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complexity ofDd-CF-FVS, andP≤3??-CF-OCTobtain the following results.
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1. Dd-CF-FVSadmits aO(kO(d)) kernel.
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2. P≤3??-CF-OCTdoes not admit polynomial kernel, unlessNP⊆ coNPpoly.
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Note that D0 denotes edgeless graphs and henceD0-CF-FVS, andD0-CF-OCT are
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essentially FVS, andOCT, respectively. Thus, any polynomial kernel forDd-CF-FVS, and
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P≤3??-CF-OCT, must generalize the known kernels for these problems. We remark that the
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above result imply thatCF-FVSadmits polynomial kernels, when the conflict graph belong
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to several well studied graph families, such as planar graphs, graphs of bounded degree, graphs
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of bounded treewidth, graphs excluding some fixed graph as a minor, a topological minor
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and graphs of bounded expansion etc. (all these graphs classes have bounded degeneracy).
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Strategy for CF-FVS.Our kernelization algorithm forCF-FVSconsists of the following
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two steps. The first step of our kernelization algorithm is a structural decomposition of the
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input graphG. This does not depend on the conflict graphH. In this phase of the algorithm,
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given an instance (G, H, k) of CF-FVS we obtain an equivalent instance (G0, H0, k0) of
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CF-FVSsuch that:
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The minimum degree ofG0 is at least 2.
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The number of vertices of degree at least 3 inG0 is upper bounded byO(k3). LetV≥3
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denote the set of vertices of degree at least 3 inG0.
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The number of maximal degree 2 paths in G0 is upper bounded by O(k3). That is,
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G0−V≥3 consists ofO(k3) connected components where each component is a path.
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We obtain this structural decomposition using reduction rules inspired by the quadratic
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kernel forFVS[25]. As stated earlier, this step can be performed for any graphH. Thus the
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problem reduces to designing reduction rules that bound the number of vertices of degree 2
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in the reduced graph. Note that we can not do this for any arbitrary graphH as the problem
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is W[1]-hard. Once the decomposition is obtained we can not use the knownreduction rules
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forFVS. This is for a simple reason that inG0 the only vertices that are not bounded have
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degree exactly 2 inG0. On the other hand forFVSwe can do simple “short-circuit” of degree
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2 vertices (remove the vertex and add an edge between its two neighbors) and assume that
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there is no vertices of degree two in the graph. So our actual contributions start here.
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The second step of our kernelization algorithm bounds the degree two vertices in the
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graph G0. Here we must use the properties of the graphH. We propose new reduction
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rules for bounding degree two vertices, whenH belongs to the family ofd-degenerate graphs.
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Towards this we use the notion ofd-degeneracy sequence, which is an ordering of the vertices
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inH such that any vertex can have at mostdforward neighbors. This is used in designing a
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marking scheme for the degree two vertices. Broadly speaking our marking scheme associates
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a set with every vertexv. Here, set consists of “ paths and cycles ofG0 on which the forward
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neighbors of v are”. Two vertices are called similar if their associated sets are same. We
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show that if some vertex is not marked then we can safely contract this vertex to one of its
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neighbors. We then upper bound the degree two vertices byO(kO(d)dO(d)), and thus obtain
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a kernel of this size forDd-CF-FVS.
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At the heart of our kernelization algorithm is a combinatorial tool of “k-independence
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preserver”. Informally, it is a set of “important” vertices for a given subsetX ⊆V(H), that
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is enough to capture the independent set property inH. We show that for d-degenerate
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graph independence preserver of sizekO(d)exists, and can be used in designing polynomial
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kernel. This is our main conceptual contribution.
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Strategy for CF-OCT.The kernelization lower bound is obtained by the method of
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cross-composition [6]. We first define a conflict version of thes-t-Cutproblem, whereH
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belongs toP≤3??. Then, we show that the problem is NP-hard and cross composes to itself.
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Finally, we give a parameter preserving reduction from the problem to P≤3??-CF-OCT, and
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obtain the desired kernel lower bound.
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Related Work. In the past, the conflict free versions of some classical problems have
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been studied, e.g. forShortest Path[15], Maximum Flow[21, 22],Knapsack [23],Bin
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Packing[12],Scheduling[13],Maximum MatchingandMinimum Weight Spanning
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Tree[10, 9]. It is interesting to note that some of these problems areNP-hard even when
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their non-conflicting version is polynomial time solvable. The study of conflict free problems
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has also been recently initiated in computational geometry motivated by various applications
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(see [1, 2, 3]).
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2 Preliminaries
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Throughout the paper, we follow the following notions. LetGbe a graph,V(G) andE(G)
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denote the vertex set and the edge set of graphG, respectively. Letn andmdenote the
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number of vertices and the number of edges of G, respectively. Let G be a graph and
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X⊆V(G), thenG[X] is the graph induced onX andG−X is graphGinduced onV(G)\X.
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Let ∆ denotes the maximum degree of graph G. We useNG(v) to denote the neighborhood
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ofv inGandNG[v] to denoteNG(v)∪ {v}. LetE0 be subset of edges of graphG, byG[E0]
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we mean the graph with the vertex setV(G) and the edge set E0. Let X ⊆E(G), then
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G−X is a graph with the vertex setV(G) and the edge set E(G)\X. LetY be a set of
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edges on vertex setV(G), thenG∪Y is graph with the vertex setV(G) and the edge set
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E(G)∪Y. Degree of a vertexv in graphGis denoted by degG(v). For an integer `, we
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denote the set{1,2, . . . , `}by [`]. ApathP ={v1, . . . , vn}is an ordered collection of vertices
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such that there is an edge between every consecutive vertices inP andv1, vn areendpoints of
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P. For a pathP byV(P) we denote set of vertices inP and byE(P) we denote set of edges
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inP. AcycleC={v1, . . . , vn}is a path with an edgev1vn. We define amaximal degree two
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induced path inGas an induced path of maximal length such that all vertices in path are of
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degree exactly two inG. Anisolated cyclein graphGis defined as an induced cycle whose
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all the vertices are of degree exactly two inG. LetG0 andGbe graphs,V(G0)⊆V(G) and
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E(G0)⊆E(G), then we say thatG0 is asubgraph of G. The subscript in the notations will
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be omitted if it is clear from the context.
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A graphGhasdegeneracydif every subgraph ofGhas a vertex of degree at mostd. An
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ordering of verticesσ:V(G)→ {1,· · ·, n}is is called ad-degeneracy sequenceof graphG, if
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every vertexv has at mostdneighborsuwithσ(u)> σ(v). A graphGisd-degenerate if and
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only if it has ad-degeneracy sequence. For a vertexvind-degenerate graphG, the neighbors
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of v which comesafter (before)v ind-degeneracy sequence are calledforward (backward)
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neighborsofvin the graphG. Given ad-degenerate graph, we can findd-degeneracy sequence
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in linear time [18].
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3 A Tool for Our Kernelization Algorithm
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In this section, we give a tool, which we believe might be useful in obtaining kernelization
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algorithm for “conflict free” versions of various parameterized problems (admitting kernels),
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when the conflict graph belongs to the family ofd-degenerate graphs. We particularly use
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this tool to obtain kernel for Dd-CF-FVS (Section 4). For a parameterized problem Π,
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consider an instance (G, H, k) of its conflict free variant,Conflict FreeΠ. Then in the
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kernelization step where we want to bound the number of vertices, it is seemingly useful to
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be able to obtain a set of “important” vertices for a given subsetX ⊆V(H) that will be
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enough to capture the independent set property inH. The above intuition becomes clear
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when we describe the kernelization algorithm forDd-CF-FVS.
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To formalize the notion of “important” set of vertices, we give the following definition.
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IDefinition 1. For ad-degenerate graphH and a setX ⊆V(H), ak-independence preserver
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for (H, X) is a setX0⊆X, such that for any independent setS inH of size at mostk, if
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there isv∈(S∩X)\X0, then there isv0∈X0\S, such that (S\ {v})∪ {v0}is an independent
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set inH.
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Throughout this section, we work with a (fixed)d, which is the degeneracy of the input
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graph. The goal of this section will be to obtain an algorithm for computing ak-independence
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preserver for (H, X) of “small” size. To quantify the “small” size, we need the following
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definition.
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IDefinition 2. For eachq∈[d], we define an integernq as follows.
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1. Ifq= 1, thennq =kd+k+ 1, and
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2. nq =knq−1+kd+k+ 1, otherwise.
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Next, we formally define the problem for which we want to design a polynomial time
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algorithm. We call this problemd-Bounded Independence Preserver(d-BIP, for short).
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d-Bounded Independence Preserver(d-BIP)
Input: Ad-degenerate graphH, a setX⊆V(H), and an integerk.
Output: A setX0⊆X of size at mostnd+1, such thatX0 is akindependence preserver for (H, X).
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In the following, let (H, X, k) be an instance ofd-BIP. We work with a (fixed)d-degeneracy
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sequence,σofH. We recall that such a sequence can be computed in polynomial time [18].
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Forward and backward neighbors of a vertexv are also defined with respect to the ordering
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σ. Ifσ(u)< σ(v), thenuis a backward neighbor ofv andv is a forward neighbor ofu. By
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NHf(v) (NHb(v)) we denote the set of forward (backward) neighbors of the vertexvin H.
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To design our polynomial time algorithm for d-BIP, we need the notion ofq-reducible
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sets, which is formally defined below.
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IDefinition 3. A setY ⊆V(H) is q-reducible, if for every setU ⊆Y, for which there is a
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setZ⊆V(H), such that: (i)Z is of size exactlyd−q+ 1 and (ii) for eachu∈U, we have
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Z⊆NHf(u), it holds that |U| ≤nq.
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Now, we give our polynomial time algorithm ford-BIPin Algorithm 1.
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Algorithm 1Algo1(H, X)
Require: d-degenerate graphH, X⊆V(H), and an integerk.
Ensure: X0 ⊆X of size at mostnd+1, which is a k-independence preserver of (H, X).
1: Forq∈[d], setnq =kd+ 1, whenq= 1, andnq =knq−1+kd+k+ 1, otherwise.
2: q= 1.
3: whileq≤ddo
4: while X is notq-reducibledo
5: FindU ⊆X of sizenq+ 1, for which there isZ ⊆V(H) of size exactlyd−q+ 1, such that for eachu∈U, we haveZ⊆NHf(u).
6: Letvbe an arbitrary vertex in U. 7: X =X\ {v}.
8: end while 9: q=q+ 1.
10: end while
11: while|X|> nd+1 do
12: Letv be an arbitrary vertex inX. 13: X =X\ {v}.
14: end while 15: SetX0 =X. 16: returnX0
To prove the correctness of our algorithm, we state an observation, the proof of which
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follows from the fact that any vertex can have at mostdforward neighbors inH.
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IObservation 1. LetH be ad-degenerate graph and S be an independent set ofH of size
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at mostk. Then, for any setU ⊆V(H), such that for each vertexu∈U,NHb(u)∩S6=∅,
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we have that|U| ≤kd.
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Now we are ready to prove the correctness of our algorithm (Algorithm 1) ford-BIP.
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ILemma 2. Algorithm 1 is correct.
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Proof. Let (H, X, k) be an instance ofd-BIP, andX0 be the output returned by Algorithm 1
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with it as the input. Clearly,X0⊆X as we do not add any new vertex to obtain the setX0,
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and size of X0 is bounded bynd+1, since at Step 10-13 of the algorithm we reduce its size
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to (at most)nd+1. Therefore, it remains to show thatX0 is ak-independence preserver of
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(H, X). To this end, we consider the following cases.
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Case 1: X isq-reducible, for eachq∈[d]. In this case, the algorithm arbitrarily deletes
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vertices (if required) from X to obtain X0. If X = X0, then the claim trivially holds.
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Therefore, we assume thatX0 is a strict subset ofX. To show thatX0 is a k-independence
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preserver for (H, X), consider an independent setS inH of size at mostk. Furthermore,
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consider a vertexv∈(S∩X)\X0 (again, if such a vertex does not exists, the claim follows).
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To prove the desired result, we want to find a replacement vertex forvinX0 which can be
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added toS (after removing v) to obtain an independent set inH. To this end, we mark
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some vertices in X0. Firstly, mark all the forward neighbors of each s∈S in the setX0.
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That is, we letXM0 to be the set (∪s∈SNHf(s))∩X0. Also, we add all vertices inS∩X0
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to the set XM0 . By the property of d-degeneracy sequence, we have that |XM0 | ≤ kd+k
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(see Observation 1). Next, we will mark some more vertices inXM0 with the hope to find
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a replacement vertex forv inX0\XM0 to add toS. Recall that by our assumption X is
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q-reducible, for each q∈[d], and in particular, it isd-reducible. Thus, for eachs∈S, the
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setXs={x∈X |s∈NHf(x)} ⊆X has size at mostnd. Based on the above observation,
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we describe our second level of marking of vertices inX0. For each s ∈ S, we add each
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vertex inXstoXM0 . From the discussions above, we have that|XM0 | ≤kd+k+knd. Since
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|X0|=nd+1, and by definition,nd+1=knd+kd+k+ 1, we haveX0\XM0 6=∅. Moreover,
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no vertex in X0 has a neighbor in S\ {v}. Therefore, for v0 ∈ X0\XM0 , we have that
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S0= (S\ {v})∪ {v0}is an independent set inH.
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Case 2: X is notq-reducible, for someq∈[d]. Letq0 be the smallest integer for which
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X is notq0-reducible. SinceX is notq0-reducible, there is a setU ⊆X of size at leastnq+ 1,
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for which there is a setZ ⊆V(H) of size exactlyd−q+ 1, such that for each u∈U, we
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haveZ⊆NHf(u). Consider (first) such pair of setsU, Z considered by the algorithm in Step
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4. Furthermore, letv∈U be the vertex deleted by the algorithm in Step 6. Let ˆU =U\ {v}.
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To prove the claim, it is enough to show that for an independent setS of size at most k
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containing v inH, there isv0 ∈Uˆ such that (S\ {v})∪ {v0} is an independent set in H.
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Here, we will use the fact that deleting a vertex from a set does not change a set from being
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˜
q-reducible to a set which is not ˜q-reducible, where ˜q ∈[d]. In the following, consider an
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independent set S of size at mostk containing v inH. We construct a marked set ˆUM,
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of vertices in ˆU. Firstly, we add all the vertices in (∪s∈S\{v}NHf(s))∩Uˆ to ˆUM. Also, we
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add all vertices inS∩Uˆ to ˆUM. Notice that at the end of above marking scheme, we have
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|XˆM| ≤kd+k. We will mark some more vertices in ˆU. Before stating the second level of
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marking, we remark thatS∩Z=∅. For eachs∈S\ {v}, letZs=Z∪ {s}. SinceS∩Z=∅,
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we have that|Zs|=d−(q−1) + 1. Fors∈S\ {v}, let ˆUs={u∈Uˆ |Zs⊆NHf(u)}. SinceX
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isq∗-reducible for eachq∗< q0, we have|Uˆs| ≤nq−1, for eachs∈S\ {v}. Now we are ready
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to describe our second level of marking. For eachs∈S\ {v}, add all vertices inUsto the set
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UˆM. Notice that|UˆM| ≤kd+k+knq−1. Moreover,|Uˆ| ≥nq andnq=knq−1+kd+k+ 1.
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Thus, there is a vertexv0 ∈Uˆ\UˆM, such that (S\ {v})∪ {v0}is an independent set inH. J
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ILemma 3. (?)2 Algorithm 1 runs in timenO(d).
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Using Lemma 2 and Lemma 3 we obtain the following theorem.
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2 The proofs of results marked with?will appear in the full version of the paper.
ITheorem 4. d-Bounded Independence Preserver admits an algorithm running in
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timenO(d).
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4 A Polynomial Kernel for D
d-CF-FVS
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In this section, we design a kernelization algorithm forDd-CF-FVS.
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To design a kernelization algorithm forDd-CF-FVS, we define another problem calledDd-
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Disjoint-CF-FVS(Dd-DCF-FVS, for short). We first define the problemDd-DCF-FVS
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formally, and then explain its uses in our kernelization algorithm.
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Dd-Disjoint-CF-FVS(Dd-DCF-FVS) Parameter: k Input: An undirected graph G, a graph H ∈ Dd such that V(G) =V(H), a subset R⊆V(G), and a non-negative integerk.
Question: Is there a setS⊆V(G)\Rof size at mostk, such thatG−Sdoes not have any cycle andS is an independent set inH?
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Notice thatDd-CF-FVS is a special case of Dd-DCF-FVS, whereR = ∅. Given an
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instance ofDd-CF-FVS, the kernelization algorithm creates an instance ofDd-DCF-FVS
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by settingR=∅. Then it applies a kernelization algorithm forDd-DCF-FVS. Finally, the
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algorithm takes the instance returned by the kernelization algorithm forDd-DCF-FVSand
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generates an instance ofDd-CF-FVS. Before moving forward, we note that the purpose
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of having set R is to be able to prohibit certain vertices to belong to a solution. This is
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particularly useful in maintaining the independent set property of the solution, when applying
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reduction rules which remove vertices from the graph (with an intention of it being in a
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solution).
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We first focus on designing a kernelization algorithm forDd-DCF-FVS, and then give
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a polynomial time linear parameter preserving reduction fromDd-DCF-FVStoDd-CF-
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FVS. If the kernelization algorithm forDd-DCF-FVSreturns that (G, H, R, k) is aYES
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(NO) instance ofDd-DCF-FVS, then conclude that (G, H, k) is aYES(NO) instance of
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Dd-CF-FVS. In the following, we describe a kernelization algorithm forDd-DCF-FVS. Let
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(G, H, R, k) be an instance ofDd-DCF-FVS. The algorithm starts by applying the following
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simple reduction rules.
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IReduction Rule 1.
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(a) Ifk≥0 andGis acyclic, then return that (G, H, R, k) is aYESinstance ofDd-DCF-
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FVS.
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(b) Return that (G, H, R, k) is a NO instance of Dd-DCF-FVS, if one of the following
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conditions is satisfied:
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(i) k≤0 andGis not acyclic,
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(ii) Gis not acyclic andV(G)⊆R, or
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(iii) There are more than kisolated cycles inG.
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IReduction Rule 2.
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(a) Letv be a vertex of degree at most 1 inG. Then deletev from the graphsG, H and the
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setR.
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(b) If there is an edge inG(H) with multiplicity more than 2 (more than 1), then reduce
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its multiplicity to 2 (1).
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(c) If there is a vertex v with self loop in G. Ifv /∈R, delete v from the graphs Gand
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H, and decrease k by one. Furthermore, add all the vertices inNH(v) to the set R,
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otherwise return that (G, H, R, k) is aNOinstance ofDd-DCF-FVS.
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(d) If there are parallel edges between (distinct) verticesu, v∈V(G) inG:
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(i) Ifu, v∈R, then return that (G, H, R, k) is aNOinstance ofDd-DCF-FVS.
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(ii) Ifu∈R (v∈R), delete v (u) from the graphsGandH, and decreasek by one.
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Furthermore, add all the vertices inNH(v) (NH(u)) to the setR.
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It is easy to see that the above reduction rules are correct, and can be applied in
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polynomial time. In the following, we define some notion and state some known results,
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which will be helpful in designing our next reduction rules.
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IDefinition 4. For a graphG, a vertexv∈V(G), and an integert∈N, at-flower atv is a
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set oft vertex disjoint cycles whose pairwise intersection is exactly{v}.
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IProposition 1. [8, 19, 25] For a graphG, a vertexv∈V(G) without a self-loop inG, and
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an integerk, the following conditions hold.
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(i) There is a polynomial time algorithm, which either outputs a (k+ 1)-flower atv, or it
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correctly concludes that no such (k+1)-flower exists. Moreover, if there is no (k+1)-flower
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at v, it outputs a setXv⊆V(G)\ {v} of size at most 2k, such thatXv intersects every
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cycle passing throughv in G.
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(ii) If there is no (k+ 1)-flower at v in Gand the degree of v is at least 4k+ (k+ 2)2k.
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Then using a polynomial time algorithm we can obtain a setXv ⊆V(G)\ {v} and a
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set Cv of components ofG[V(G)\(Xv∪ {v})], such that each component inCv is a tree,
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v has exactly one neighbor in C∈ Cv, and there exist at leastk+ 2 components inCv
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corresponding to each vertex x∈Xv such that these components are pairwise disjoint
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and vertices inXv have an edge to each of their associated components.
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IReduction Rule 3. Consider v∈V(G), such that there is a (k+ 1)-flower atv inG. If
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v∈R, then return that (G, H, R, k) is aNOinstance ofDd-DCF-FVS. Otherwise, deletev
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fromG, H and decrease kby one. Furthermore, add all the vertices inNH(v) toR.
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The correctness of the above reduction rule follows from the fact that such a vertex must
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be part of every solution of size at mostk. Moreover, the applicability of it in polynomial
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time follows from Proposition 1 (item (i)).
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I Reduction Rule 4. Let v ∈ V(G), Xv ⊆V(G)\ {v}, and Cv be the set of components
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which satisfy the conditions in Proposition 1(ii) (inG), then delete edges betweenv and the
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components of the setCv, and add parallel edges betweenv and every vertexx∈Xv in G.
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The polynomial time applicability of Reduction Rule 4 follows from Proposition 1. And,
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in the following lemma, we prove the safeness of this reduction rule.
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ILemma 5. (?)Reduction Rule 4 is safe.
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In the following, we state an easy observation, which follows from non-applicability of
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Reduction Rule 1 to 4.
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IObservation 6. Let (G, H, R, k) be an instance ofDd-DCF-FVS, where none of Reduction
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Rule 1 to 4 apply. Then the degree of each vertex inGis bounded byO(k2).
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Proof. As Reduction Rule 3 is not applicable, then there is nok+ 1-flower inG. Now, if
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there is v ∈ V(G) with degree at least 4k+ (k+ 2)2k, then Reduction Rule 4 would be
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applicable. J
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To design our next reduction rule, we construct an auxiliary graph G?. Intuitively
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speaking, G? is obtained from Gby shortcutting all degree two vertices. That is, vertex
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set ofG? comprises of all the vertices of degree at least three in 3. From now on, vertices
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of degree at least 3 (inG) will be referred to as high degree vertices. For eachuv∈E(G),
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whereu, v are high degree vertices, we add the edgeuvinG?. Furthermore, for an induced
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maximal pathPuv, betweenuandv where all the internal vertices of Puv are degree two
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vertices inG, we add the (multi) edgeuvtoE(G?). Next, we will use the following result to
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bound the number of vertices and edges inG?.
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IProposition 2. [8] A graphGwith minimum degree at least 3, maximum degree ∆, and a
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feedback vertex set of size at mostk has at most (∆ + 1)kvertices and 2∆k edges.
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The above result (together with the construction of G?) gives us the following (safe)
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reduction rule.
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IReduction Rule 5. If|V(G?)| ≥4k2+ 2k2(k+ 2) or|E(G?)| ≥8k2+ 4k2(k+ 2), then return
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NO.
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ILemma 7. Let(G, H, R, k)be an instance ofDd-DCF-FVS, where none of the Reduction
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Rules 1 to 5 are applicable. Then we obtain the following bounds:
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The number of vertices of degree at least3 in Gis bounded byO(k3).
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The number of maximal degree two induced paths inGis bounded byO(k3).
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Having shown the above bounds, it remains to bound the number of degree two vertices
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inG. We start by applying the following simple reduction rule to eliminate vertices of degree
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two inG, which are also inR.
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IReduction Rule 6. Letv ∈R, dG(v) = 2, and x, y be the neighbors ofv in G. Delete v
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from the graphsG, H and the setR. Furthermore, add the edgexyin G.
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The correctness of this reduction rule follows from the fact that vertices inRcan not be part
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of any solution and all the cycles passing throughvalso passes through its neighbors.
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In the polynomial kernel for the Feedback Vertex Set problem (with no conflict
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constraints), we can short-circuit degree two vertices. But in our case, we cannot perform
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this operation, since we also need the solution to be an independent set in the conflict
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graph. Thus to reduce the number of degree two vertices inG, we exploit the properties
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of ad-degenerate graph. To this end, we use the tool that we developed in Section 3. This
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immediately gives us the following reduction rule.
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IReduction Rule 7. LetP be a maximal degree two induced path inG. If|V(P)| ≥nd+1+ 1,
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apply Algorithm 1 with input (H, V(P)\R). LetVb(P) be the set returned by Algorithm 1.
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Letv∈(V(P)\R)\Vb(P), andx, ybe the neighbors ofv inG. Deletev from the graphs
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G, H. Furthermore, add edgexy inG.
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ILemma 8. Reduction Rule 7 is safe.
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Proof. Let (G, H, R, k) be an instance of Dd-DCF-FVSandv be a vertex in a maximal
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degree two pathP with neighborsxandy, with respect to which Reduction Rule 8 is applied.
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Furthermore, let (G0, H0, R, k) be the resulting instance after application of the reduction
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rule. We will show that (G, H, R, k) is a YES instance of Dd-DCF-FVS if and only if
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(G0, H0, R, k) is aYES instance ofDd-DCF-FVS.
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In the forward direction, let (G, H, R, k) be aYESinstance ofDd-DCF-FVS andS be
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one of its minimal solution. Consider the case whenv /∈S. In this case, we claim that S
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is also a solution ofDd-DCF-FVSfor (G0, H0, R, k). Suppose not then eitherS is not an
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independent set inH0 orG0−Scontains a cycle. Since,H0 is an induced subgraph ofH, we
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have thatS0 is also an independent set inH0. So we assume thatG0−S has a cycle, sayC.
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IfC does not contain the edgexy, thenC is also a cycle inG−S. Therefore, we assume
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thatC contains the edgexy. But then (C\ {xy})∪ {xv, vy} is a cycle inG−S. Next, we
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consider the case whenv ∈S. By Lemma 2 we have a vertex v0 ∈ V(P)\ {v} such that
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(S\ {v})∪ {v0}is an independent set inH0. By using the fact that any cycle that passes
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throughvalso contains all vertices in P (together with the discussions above) imply that
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(S\ {v})∪ {v0}is a solution ofDd-DCF-FVSfor (G0, H0, R, k).
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In the reverse direction, let (G0, H0, R, k) be aYESinstance ofDd-DCF-FVSandS0
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be one of its minimal solution. We claim thatS0 is also a solution of Dd-DCF-FVSfor
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(G, H, R, k). Suppose not, then eitherS is not an independent set inH orG−S contains a
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cycle. Since,H0 is an induced subgraph ofH, we have thatS0is also an independent set inH.
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Next, assume that there is a cycleC inG−S. The cycleCmust containv, otherwise,Cis
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also a cycle inG0−S0. Sincev is a degree two vertex inG, therefore any cycle that contains
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v, must also containxandy. As observed before, G− {xv, vy} is identical to G0− {xy}.
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But then, (C\ {xv, vy})∪ {xy}is a cycle inG0−S0, a contradiction. This concludes thatS0
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is a solution ofDd-DCF-FVSfor (G, H, R, k). J
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I Lemma 9. (?) Let (G, H, R, k) be an instance of Dd-DCF-FVS, where none of the
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Reduction Rules 1 to 7 are applicable. Then the number of vertices in a degree two induced
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path in Gis bounded byO(kO(d)).
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ITheorem 10. Dd-DCF-FVS admits a kernel with O(kO(d))vertices.
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ILemma 11. (?)There is a polynomial time parameter preserving reduction fromDd-DCF-
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FVStoDd-CF-FVS.
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By Theorem 10 and Lemma 11, we obtain the following result.
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ITheorem 12. Dd-CF-FVS admits a kernel withO(kO(d))vertices.
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5 Kernelization Complexity of P
≤3??- CF-OCT
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In this section, we show thatCF-OCTdoes not admit a polynomial kernel when the conflict
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graph belongs to the familyP≤3??. LetP≤3 denotes the family of disjoint union of paths of
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length at most three, andP≤3? denotes the family of disjoint union of paths of length at most
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three and a star graph. We give parameter preserving reduction fromP≤3? -Conflict Free
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s-t Cut(P≤3? -CF-s-t Cut) toP≤3??-CF-OCT.
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We first prove that P≤3? -CF-s-t Cut is NP-hard. Then, we prove thatP≤3? -CF-s-t
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Cut does not admit a polynomial compression, unless NP ⊆ coNPpoly using the method of
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cross-composition.
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I Theorem 13 (?). P≤3? -CF-s-t Cut does not admit a polynomial compression unless
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NP⊆ coNPpoly.
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Lower Bound for Kernel of P≤3??-CF-OCT.In this subsection, we prove the main
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result of this section. We show that there does not exist a polynomial kernel of P≤3??-
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CF-OCT. Towards this we give a parameter preserving reduction fromP≤3? -CF-s-t Cut
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toP≤3??-CF-OCT. Given an instance (G, H, s, t, k) of P≤3? -CF-s-t Cut, we construct an
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instance (G0, H0, k+ 1) of P≤3??-CF-OCT as follows. Initially, we haveV(G0) =V(H0) =
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V(G)∪ {z, a, b}. Now, for each edgeei∈E(G), add a vertexwi toV(G0) andV(H0). Now,
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we define the edge set ofG0. Letxi, yi be end points ofei∈E(G). For eachei∈E(G), add
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edgesxiwi andyiwi toE(G0). Also, add a self loop onz inG0 and edges sa, abandbtto
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E(G0). To construct the edge set ofH0, we setE(H0) =E(H− {s, t}). Additionally, we add
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zs, zt, za, zt, andzwifor each wi ∈V(H0) toE(H0). Figure 1 describes the construction of
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G0 andH0.
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V(G)
GraphG0 GraphH0
H xi
yi xj yj
s
t
wi
wj z
a
b
... ... ......
z a b
w1 w2
...
w|E(G)|
Figure 1An illustration of construction of graphG0andH0 in reduction fromP≤3? -CF-s-tCut toP≤3??-CF-OCT.
Clearly,H0 belongs toP3?? and this construction can be carried out in the polynomial
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time. Now, we prove the equivalence between the instances (G, H, s, t, k) ofP≤3? -CF-s-t
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Cutand (G0, H0, k+ 1) ofP≤3??-CF-OCTin the following lemma.
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ILemma 14. (G, H, s, t, k)is a yes-instance ofP≤3? -CF-s-tCutif and only if(G0, H0, k+1)
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is a yes-instance ofP≤3??-CF-OCT.
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Proof. In the forward direction, let (G, H, s, t, k) be a yes-instance ofP≤3? -CF-s-t Cut
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andS be one of its solution. We claim that S∪ {z} is a solution toP≤3??-CF-OCT in
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(G0, H0, k+ 1). In the graphG0, since we subdivide each edge, all the paths froms−t are of
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even length. Since, we subdivide each edge ofG,G0− {a, b, z} is a bipartite graph. Hence,
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an odd cycle inG0−zconsists of ans−tpath inG0− {a, b}and edgessa,abandbt. Clearly,
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by the construction ofG0, (G0− {a, b})\S does not contain ans−tpath and henceG0−z
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does not contain an odd cycle. Since,H[S] is edgeless,S∪ {z} is an independent set inH0.
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This completes the proof in the forward direction.
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In the reverse direction, letS be a solution toP≤3??-CF-OCTin (G0, H0, k+ 1). Since,
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z ∈ S, therefore, s, t, a, b, wi ∈/ S for any wi ∈ V(H0). We claim that S0 = S \ {z}
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is a solution to P≤3? -CF-s-t Cut in (G, H, s, t, k). Suppose not, then there exists a
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s−t path (s, x1, x2,· · · , xl, t) in G\ S0. Correspondingly, there exists a s−t path
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(s, w1, x1, w2, x2,· · · , xl, wl+1, t) in G0 of even length which results into an odd cycle
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(s, w1, x1, w2, x2,· · · , xl, wl+1, t, b, a) inG0\S, a contradiction. This completes the proof.
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J
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Now, we present the main result of this section in the following theorem.
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ITheorem 15. P≤3??-CF-OCTdoes not admit a polynomial kernel. unlessNP⊆ coNPpoly.
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6 Conclusion
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In this paper we studied kernelization complexity ofDd-CF-FVS andDd-CF-OCT. We
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showed that the former admits a polynomial kernel of sizekO(d), whileDd-CF-OCTdoes not
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admit any polynomial kernel unlessNP⊆ coNPpoly. In fact, the later does not admit polynomial
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kernel even for much more specialized problem, namelyP≤3??-CF-OCT. Using much more
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involved marking scheme we can show thatDd-CF-ECT admits polynomial kernel of size
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kO(d). Similarly, we can extend the known polynomial kernel forOCTtoCF-OCTwhen
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the conflict graphH has maximum degree at most one. Two most interesting questions that
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still remain open form our work are following: (a) doesCF-FVSadmit uniform polynomial
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kernel on graphs of bounded expansion; and (b) doesCF-OCTadmit a polynomial kernel
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whenH is disjoint union of paths of length at most 2.
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