Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity
Tatsuya Gima, Tesshu Hanaka, Masashi Kiyomi, Yasuaki Kobayashi, Yota Otachi
EExploring the Gap Between Treedepth andVertex Cover Through Vertex Integrity (cid:63)
Tatsuya Gima , Tesshu Hanaka , Masashi Kiyomi , Yasuaki Kobayashi , andYota Otachi Nagoya University, Nagoya, Japan [email protected] , [email protected] Chuo University, Bunkyo-ku, Tokyo, Japan [email protected] Yokohama City University, Yokohama, Japan [email protected] Kyoto University, Kyoto, Japan [email protected]
Abstract.
For intractable problems on graphs of bounded treewidth,two graph parameters treedepth and vertex cover number have beenused to obtain fine-grained complexity results. Although the studies inthis direction are successful, we still need a systematic way for furtherinvestigations because the graphs of bounded vertex cover number forma rather small subclass of the graphs of bounded treedepth. To fill thisgap, we use vertex integrity, which is placed between the two parame-ters mentioned above. For several graph problems, we generalize fixed-parameter tractability results parameterized by vertex cover number tothe ones parameterized by vertex integrity. We also show some finer com-plexity contrasts by showing hardness with respect to vertex integrity ortreedepth.
Keywords: vertex integrity, vertex cover number, treedepth.
Treewidth, which measures how close a graph is to a tree, is arguably one of themost powerful tools for designing efficient algorithms for graph problems. Theapplication of treewidth is quite wide and the general theory built there oftengives a very efficient algorithm (e.g., [10,2,17]). However, still many problems arefound to be intractable on graphs of bounded treewidth (e.g., [50]). To cope withsuch problems, one may use pathwidth, which is always larger than or equal totreewidth. Unfortunately, this approach did not quite work as no natural problemwas known to change its complexity with respect to treewidth and pathwidth,until very recently [8]. Treedepth is a further restriction of pathwidth. How-ever, still most of the problems do not change their complexity, except for some (cid:63)
Partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP18K11168,JP18K11169, JP19K21537, JP20K19742, JP20H05793. a r X i v : . [ c s . D S ] J a n Gima et al. problems with hardness depending on the existence of long paths (e.g., [24,39]).One successful approach in this direction is parameterization by the vertex covernumber, which is a strong restriction of treedepth. Many problems that are in-tractable parameterized by treewidth have been shown to become tractable whenparameterized by vertex cover number [27,25,28,1,41,14].One drawback of the vertex-cover parameterization is its limitation to a verysmall class of graphs. To overcome the drawback, we propose a new approach forparameterizing graph problems by vertex integrity [5]. The vertex integrity of agraph G , denoted vi ( G ), is the minimum integer k satisfying that there is S ⊆ V ( G ) such that | S | + | V ( C ) | ≤ k for each component C of G − S . We call such S a vi ( k ) -set of G . This parameter is bounded from above by vertex cover number +1 and from below by treedepth. As a structural parameter in parameterizedalgorithms, vertex integrity (and its close variants) was used only in a coupleof previous studies [23,33,12]. Our goal is to fill some gaps between treedepthand vertex cover number by presenting finer algorithmic and complexity resultsparameterized by vertex integrity. Note that the parameterization by vertexintegrity is equivalent to the one by (cid:96) -component order connectivity + (cid:96) [22]. Short preliminaries.
For the basic terms and concepts in the parameterized com-plexity theory, we refer the readers to standard textbooks, e.g. [21,19].For a graph G , we denote its treewidth by tw ( G ), pathwidth by pw ( G ),treedepth by td ( G ), and vertex cover number by vc ( G ). (See Section A for defi-nitions.) It is known that tw ( G ) ≤ pw ( G ) ≤ td ( G ) − ≤ vi ( G ) − ≤ vc ( G ) forevery graph G . We say informally that a problem is fixed-parameter tractable“parameterized by vi ”, which means “parameterized by the vertex integrity ofthe input graphs.” We also say “graphs of vi = c (or vi ≤ c )”. Our results.
The main contribution of this paper is to generalize several knownFPT algorithms parameterized by vc to the ones by vi . We also show someresults considering parameterizations by vc , vi , or td to tighten the complexitygaps between parameterizations by vc and by td . See Table 1 for the summaryof results. Due to the space limitation, we had to move most of the results intothe appendix. In the main text, we present full descriptions of selected resultsonly. (Even for the selected results, we still have to omit some proofs. They aremarked with (cid:70) .) Extending FPT results parameterized by vc . We show that
Imbalance , Max-imum Common (Induced) Subgraph , Capacitated Vertex Cover , Ca-pacitated Dominating Set , Precoloring Extension , Equitable Col-oring , and
Equitable Connected Partition are fixed-parameter tractableparameterized by vertex integrity. We present the algorithms for
Imbalance asa simple but still powerful example that generalizes known results (Section 2)and for
Maximum Common Subgraph as one of the most involved examples(Section 3). See Section E for the other problems. A commonly used trick is toreduce the problem instance to a number of instances of integer linear program-ming, while each problem requires a nontrivially tailored reduction dependingon its structure. It was the same for parameterizations by vc , but the reductionshere are more involved because of the generality of vi . Finding the similarity reedepth, Vertex Cover, and Vertex Integrity 3 Table 1.
Summary. The results stated without references are shown in this paper.
Problem
Lower bounds Upper bounds
Imbalance
NP-h [9] FPT by tw + ∆ [42]FPT by vi Max Common Subgraph
NP-h for vi ( G ) = 3 FPT by vi ( G ) + vi ( G ) Max Common Ind. Subgraph
NP-h for vc ( G ) = 0 Capacitated Vertex Cover
W[1]-h by td [20] FPT by vi Capacitated Dominating Set
W[1]-h by td + k [20] FPT by vi Precoloring Extension
W[1]-h by td [26] FPT by vi Equitable Coloring
W[1]-h by td [26] FPT by vi Equitable Connected Part.
W[1]-h by pw [25] FPT by vi Bandwidth
W[1]-h by td FPT by vc [27]NP-h for pw = 2 [46] P for pw ≤ Graph Motif
NP-h for vi = 4 FPT by vc [14]P for vi ≤ Steiner Forest
NP-h for vi = 5 [35] XP by vc Unweighted Steiner Forest
NP-h for tw = 3 [35] FPT by vc Unary Min Max Outdeg. Ori.
W[1]-h by vc XP by tw [51] Binary Min Max Outdeg. Ori.
NP-h for vc = 3 P for vc ≤ Metric Dimension
W[1]-h by pw [13] FPT by tw + ∆ [6]FPT by td Directed ( p, q ) -Edge Dom. Set W[1]-h by pw [7] FPT by tw + p + q [7]FPT by td List Hamiltonian Path
W[1]-h by pw [43] FPT by td among the reductions and algorithms would be a good starting point to developa general way for handling problems parameterized by vi (or vc ). Additionally,we show that Bandwidth is W[1]-hard parameterized by td , while we were notable to extend the algorithm parameterized by vc to the one by vi . Filling some complexity gaps.
We observe that
Graph Motif and
SteinerForest have different complexity with respect to vc and vi (Section F). In par-ticular, we see that not all FPT algorithms parameterized by vc can be general-ized to the ones by vi . Min Max Outdegree Orientation gives an examplethat a known hardness for td can be strengthened to the one for vc (Section 4).We additionally observe that some W[1]-hard problems parameterized by tw become tractable parameterized by td . Such problems include Metric Dimen-sion , Directed ( p, q )- Edge Dominating Set , and
List Hamiltonian Path (Section G).
In this section, we show that
Imbalance is fixed-parameter tractable param-eterized by vi . Let G = ( V, E ) be a graph. Given a linear ordering σ on V ,the imbalance im σ ( v ) of v ∈ V is the absolute difference of the numbers of theneighbors of v that appear before v and after v in σ . The imbalance of G , de- Gima et al. noted im ( G ), is defined as min σ (cid:80) v ∈ V im ( v ), where the minimum is taken overall linear orderings on V . Given a graph G and an integer b , Imbalance askswhether im ( G ) ≤ b .Fellows et al. [27] showed that Imbalance is fixed-parameter tractable pa-rameterized by vc . Recently, Misra and Mittal [44] have extended the result byshowing that Imbalance is fixed-parameter tractable parameterized by the sumof the twin-cover number and the maximum twin-class size. Although twin-covernumber is incomparable with vertex integrity, the combined parameter in [44]is always larger than or equal to the vertex integrity of the same graph. On theother hand, the combined parameter can be arbitrarily large for some graphs ofconstant vertex integrity (e.g., disjoint unions of P ’s). Hence, our result hereproperly extends the result in [44] as well. Key concepts.
Before proceeding to the algorithm, we need to introduce twoimportant concepts that are common in our algorithms parameterized by vi .1. ILP parameterized by the number of variables.
It is known that the fea-sibility of an instance of integer linear programming (ILP) parameterized bythe number of variables is fixed-parameter tractable [40]. Using the algorithmfor the feasibility problem as a black box, one can show the same fact for theoptimization version as well. (See Section B for the detail.) This fact has beenused heavily for designing FPT algorithms parameterized by vc (see e.g. [27]).We are going to see that some of these algorithms can be generalized for theparameterization by vi , and Imbalance is the first such example.2.
Equivalence relation among components.
For a vertex set S of G , we definean equivalence relation ∼ G,S among components of G − S by setting C ∼ G,S C if and only if there is an isomorphism g from G [ S ∪ V ( C )] to G [ S ∪ V ( C )] thatfixes S ; that is, g | S is the identity function. When C ∼ G,S C , we say that C and C have the same ( G, S ) -type (or just the same type if G and S areclear from the context). See Fig. 1. We say that a component C of G − S is of( G, S ) -type t (or just type t ) by using a canonical form t of the members of the( G, S )-type equivalence class of C . We can set the canonical form t in such a waythat it can be computed from S and C in time depending only on | S ∪ V ( C ) | . Observe that if S is a vi ( k )-set of G , then the number of ∼ G,S classes dependsonly on k since | S ∪ V ( C ) | ≤ k for each component C of G − S . Hence, we cancompute for all types t the number of type- t components of G − S in O ( f ( k ) · n )total running time, where n = | V | and f ( k ) is a computable function dependingonly on k . Note that this information (the numbers of type- t components for all t ) completely characterizes the graph G up to isomorphism. Theorem 2.1.
Imbalance is fixed-parameter tractable parameterized by vi .Proof. Let S be a vi ( k )-set of G . Such a set can be found in O ( k k +1 n ) time [22].We first guess and fix the relative ordering of S in an optimal ordering. There For example, by fixing the ordering of vertices in S as v , . . . , v | S | , we can set t to be the adjacency matrix of G [ S ∪ V ( C )] such that the i th row and col-umn correspond to v i for 1 ≤ i ≤ | S | and under this condition the string t [1 , , . . . , t [1 , s ] , t [2 , , . . . , t [ s, s ] is lexicographically minimal, where s = | S ∪ V ( C ) | .reedepth, Vertex Cover, and Vertex Integrity 5 SC C C C Fig. 1.
The components C and C of G − S have the same ( G, S )-type. are only k ! candidates for this guess. For each v ∈ S , let (cid:96) ( v ) and r ( v ) be thenumbers of vertices in N ( v ) ∩ S that appear before v and after v , respectively,in the guessed relative ordering of S .Observe that the imbalance of a vertex v in a component C of G − S dependsonly on the relative ordering of S ∪ V ( C ) since N ( v ) ⊆ S ∪ V ( C ). For each type t and for each relative ordering p of S ∪ V ( C ), where C is a type- t component of G − S , we denote by im ( t, p ) the sum of imbalance of the vertices in C . Similarly,the numbers of vertices in a type- t component C that appear before v ∈ S andafter v depend only on the relative ordering p of S ∪ V ( C ); we denote thesenumbers by (cid:96) ( v, t, p ) and r ( v, t, p ), respectively. The numbers im ( t, p ), (cid:96) ( v, t, p ),and r ( v, t, p ) can be computed from their arguments in time depending only on k , and thus they are treated as constants in the following ILP.We represent by a nonnegative variable x t,p the number of type- t componentsthat have relative ordering p with S . Note that the number of combinations of t and p depends only on k . For each v ∈ S , we represent (an upper bound of) theimbalance of v by an auxiliary variable y v . This can be done by the followingconstraints: y v ≥ ( (cid:96) ( v ) + (cid:80) t,p (cid:96) ( v, t, p ) · x t,p ) − ( r ( v ) + (cid:80) t,p r ( v, t, p ) · x t,p ) ,y v ≥ ( r ( v ) + (cid:80) t,p r ( v, t, p ) · x t,p ) − ( (cid:96) ( v ) + (cid:80) t,p (cid:96) ( v, t, p ) · x t,p ) . Then the imbalance of the whole ordering, which is our objective function tominimize, can be expressed as (cid:80) v ∈ S y v + (cid:80) t,p im ( t, p ) · x t,p . Now we need the following constraints to keep the total number of type- t com-ponents right: (cid:80) p x t,p = c t for each type t, where c t is the number of components of type t in G − S .By finding an optimal solution to the ILP above for each guess of the relativeordering of S , we can find an optimal ordering. Since the number of guesses andthe number of variables depend only on k , the theorem follows. (cid:117)(cid:116) In this section, we show that
Maximum Common Subgraph (MCS) and
Max-imum Common Induced Subgraph (MCIS) are fixed-parameter tractable pa-
Gima et al. rameterized by vi of both graphs. (See Section C for the proof for MCIS.) Theresults extend known results and fill some complexity gaps as described below.A graph Q is subgraph-isomorphic to G , denoted Q (cid:22) G , if there is aninjection η from V ( Q ) to V ( G ) such that { η ( u ) , η ( v ) } ∈ E ( G ) for every { u, v } ∈ E ( Q ). A graph Q is induced subgraph-isomorphic to G , denoted Q (cid:22) I G , ifthere is an injection η from V ( Q ) to V ( G ) such that { η ( u ) , η ( v ) } ∈ E ( G ) if andonly if { u, v } ∈ E ( Q ). Given two graphs G and Q , Subgraph Isomorphism (SI) asks whether Q (cid:22) G , and Induced Subgraph Isomorphism (ISI) askswhether Q (cid:22) I G . The results of this section are on their generalizations. Giventwo graphs G and G , MCS asks to find a graph H with maximum | E ( H ) | such that H (cid:22) G and H (cid:22) G . Similarly, MCIS asks to find a graph H withmaximum | V ( H ) | such that H (cid:22) I G and H (cid:22) I G .If we restrict the structure of only one of the input graphs, then both problemsremain quite hard. Since Partition Into Triangles [34] is a special case of SIwhere the graph Q is a disjoint union of triangles, MCS is NP-hard even if one ofthe input graphs has vi = 3. Also, since Independent Set [34] is a special caseof ISI where Q is an edge-less graph, MCIS is NP-hard even if one of the inputgraphs has vc = 0. Furthermore, since SI and ISI generalize Clique [21], MCSand MCIS are W[1]-hard parameterized by the order of one of the input graphs.When parameterized by vc of one graph, an XP algorithm for (a generalizationof) MCS is known [11].For parameters restricting both input graphs, some partial results were known.It is known that SI is fixed-parameter tractable parameterized by vi of bothgraphs, while it is NP-complete when both graphs have td ≤ td . It is knownthat MCIS is fixed-parameter tractable parameterized by vc of both graphs [1]. Theorem 3.1.
Maximum Common Subgraph is fixed-parameter tractable pa-rameterized by vi of both input graphs.Proof. Let G = ( V , E ) and G = ( V , E ) be the input graphs of vertexintegrity at most k . We will find isomorphic subgraphs Γ = ( U , F ) of G and Γ = ( U , F ) of G with maximum number of edges, and an isomorphism η : U → U from Γ to Γ . Step 1. Guessing matched vi (2 k ) -sets R and R . Let S and S be vi ( k )-sets of G and G , respectively. At this point, there is no guarantee that S i ⊆ U i or η ( S ) = S . To have such assumptions, we make some guesses about η and find vi (2 k )-sets R and R of the graphs such that η ( R ) = R . Step 1-1. Guessing subsets X i , Y i ⊆ S i for i ∈ { , } . We guess disjointsubsets X and Y of S such that X = S ∩ η − ( U ∩ S ) and Y = S ∩ η − ( U \ S ). We also guess disjoint subsets X and Y of S defined similarly as X = S ∩ η ( U ∩ S ) and Y = S ∩ η ( U \ S ). Note that η ( X ) = X . Thereare 3 | S | · | S | ≤ k candidates for the combinations of X , Y , X , and Y .Observe that the vertices in S i \ ( X i ∪ Y i ) do not contribute to the isomorphicsubgraphs and can be safely removed. We denote the resultant graphs by H i . reedepth, Vertex Cover, and Vertex Integrity 7 Step 1-2. Guessing η on X ∪ Y and η − on X ∪ Y . Given the guessedsubsets X , Y , X , and Y , we further guess how η maps these subsets. There are | X | ! ≤ k ! candidates for the bijection η | X (equivalently for η − | X = ( η | X ) − ).Now we guess η | Y from at most 2 k non-isomorphic candidates as follows.Recall that η ( Y ) ⊆ V \ S . Observe that each subset A ⊆ V \ S is completelycharacterized up to isomorphism by the numbers of ways A intersects type- t components for all ( H , S )-types t . Since there are at most 2( k ) types and eachcomponent has order at most k , the total number of non-equivalent subsets ofcomponents is at most 2( k ) · k ≤ k . Since η ( Y ) is the union of at most | Y | such subsets, the number of non-isomorphic candidates of η ( Y ) is at most(2 k ) | Y | ≤ k . In the analogous way, we can guess η − | Y from at most 2 k non-isomorphic candidates.Now we set Z = η − ( Y ) and Z = η ( Y ). Let R = X ∪ Y ∪ Z and R = X ∪ Y ∪ Z . Observe that each component C of H − R satisfies that | C | ≤ k − | S | ≤ k and | C | + | R | ≤ ( k − | S | ) + ( | S | + | η − ( Y ) | ) ≤ k . Hence, R is a vi (2 k )-set of H . Similarly, we can see that R is a vi (2 k )-set of H .Furthermore, we know that η ( R ) = R . Step 2. Extending the guessed parts of η . Assuming that the guesses we made sofar are correct, we now find the entire η . Recall that we are seeking for isomorphicsubgraphs Γ = ( U , F ) of G and Γ = ( U , F ) of G with maximum numberof edges, and the isomorphism η : U → U from Γ to Γ . Since we already knowthe part η | R : R → R , it suffices to find a bijective mapping from a subset of V ( H − R ) to a subset of V ( H − R ) that maximizes the number of matchededges where the connections to R i are also taken into account.As we describe below, the subproblem we consider here can be solved byformulating it as an ILP instance with 2 O ( k ) variables. The trick here is thatinstead of directly finding the mapping, we find which vertices and edges in H i − R i are used in the common subgraph.In the following, we are going to use a generalized version of types since thevertex set of a component of H i − R i does not necessarily induce a connectedsubgraph of Γ i . It is defined in a similar way as ( H i , R i )-types except that it isdefined for each pair ( A, B ) of a connected subgraph A of H i − R i and a subset B of the edges between A and R i . Let ( A , B ) and ( A , B ) be such pairs in H i − R i . We say that ( A , B ) and ( A , B ) have the same g- ( H i , R i ) -type (orjust g-type ) if there is an isomorphism from H i ( A , B ) to H i ( A , B ) that fixes R i , where H i ( A j , B j ) is the subgraph of H i formed by B j and the edges in A j .See Fig. 2. We say that a pair ( A, B ) is of g- ( H i , R i ) -type t (or just g-type t )by using a canonical form t of the g-( H i , R i )-type equivalence class of ( A, B ).Observe that all possible canonical forms of g-types can be computed in timedepending only on k . Step 2-1. Decomposing components of H i − R i into smaller pieces. We say thatan edge { u, v } in H is used by η if u, v ∈ U and H has the edge { η ( u ) , η ( v ) } .Similarly, an edge { u, v } in H is used by η if u, v ∈ U and H has the edge { η − ( u ) , η − ( v ) } . Gima et al. R i A A B B Fig. 2.
The pairs ( A , B ) and ( A , B ) have the same g-( H i , R i )-type. Let i ∈ { , } , t be an ( H i , R i )-type, and T be a multiset of g-( H i , R i )-types.Let C be a type t component of H i − R i , C (cid:48) the subgraph of C formed by theedges used by η , and E (cid:48) the subset of the edges between C (cid:48) and R i used by η .If T coincides with the multiset of g-types of the pairs ( A, B ) such that A is acomponent of C (cid:48) and B is the subset of E (cid:48) connecting A and R i , then we saythat η decomposes the type- t component C into T .We represent by a nonnegative variable x ( i ) t,T the number of type- t componentsof H i − R i that are decomposed into T by η . We have the following constraint: (cid:80) T x ( i ) t,T = c ( i ) t for each ( H i , R i )-type t and i ∈ { , } , where the sum is taken over all possible multisets T of g-( H i , R i )-types, and c ( i ) t is the number of components of type t in H i − R i . Additionally, if there is noway to decompose a type- t component into T , we add a constraint x ( i ) t,T = 0.As each component of H i − R i has order at most k , T contains at most k elements. Since there are at most 2( k ) g-types, there are at most (2( k )) k optionsfor choosing T . Thus the number of variables x ( i ) t,T is at most 2 · k ) · (2( k )) k +1 .Now we introduce a nonnegative variable y ( i ) t that represents the number ofpairs ( A, B ) of g-type t obtained from the components of H i − R i by decomposingthem by η . The definition of y ( i ) t gives the following constraint: y ( i ) t = (cid:80) t (cid:48) , T µ ( T, t ) · x ( i ) t (cid:48) ,T for each g-( H i , R i )-type t and i ∈ { , } , where µ ( T, t ) is the multiplicity of g -type t in T and the sum is taken over allpossible ( H i , R i )-types t (cid:48) and multisets T of g-( H i , R i )-types. As in the previouscase, we can see that the number of variables y t depends only on k . Step 2-2. Matching decomposed pieces.
Observe that for each g-( H , R )-type t , there exists a unique g-( H , R )-type t such that there is an isomorphism g from H ( A , B ) to H ( A , B ) with g | R = η | R , where ( A i , B i ) is a pair ofg-( H i , R i )-type t i for i ∈ { , } . We say that such g-types t and t match . Since η is an isomorphism from Γ to Γ , η maps each g-( H , R )-type t pair to ag-( H , R )-type t pair, where t and t match. This implies that y (1) t = y (2) t ,which we add as a constraint. Now the total number of edges used by η can becomputed from y (1) t . Let m t be the number of edges in H ( A, B ), where (
A, B )is a pair of g-( H , R )-type t . Let r be the number of matched edges in R ; thatis, r = |{{ u, v } ∈ E ( H [ R ]) | { η ( u ) , η ( v ) } ∈ E ( G [ R ]) }| . Then, the number ofmatched edges is r + (cid:80) t m t · y (1) t . On the other hand, given an assignment to the reedepth, Vertex Cover, and Vertex Integrity 9 variables, it is easy to find isomorphic subgraphs with that many edges. Since r is a constant here, we set (cid:80) t m t · y (1) t to the objective function to be maximized.Since the number of candidates in the guesses we made and the number ofvariables in the ILP instances depend only on k , the theorem follows. (cid:117)(cid:116) Given an undirected graph G = ( V, E ), an edge weight function w : E → Z + ,and a positive integer r , Min Max Outdegree Orientation (MMOO) askswhether there exists an orientation Λ of G such that each vertex has outdegree atmost r under Λ , where the outdegree of a vertex is the sum of the weights of out-going edges. If each edge weight is given in binary, we call the problem BinaryMMOO , and if it is given in unary, we call the problem
Unary MMOO . Notethat in the binary version, the weight of an edge can be exponential in the inputsize, whereas the unary version does not allow such weights.
Unary MMOO admits an n O ( tw ) -time algorithm [51], but it is W[1]-hardparameterized by td [50]. In this section, we show a stronger hardness param-eterized by vc . Binary MMOO is known to be NP-complete for graphs of vi = 4 [3]. In Section D, we show a stronger hardness result that the binaryversion is NP-complete for graphs of vc = 3. This result is tight as we can showthat the binary version is polynomial-time solvable for graphs of vc ≤ Theorem 4.1.
Unary MMOO is W[1]-hard parameterized by vc .Proof. We give a parameterized reduction from
Unary Bin Packing . Givena positive integer t and n positive integers a , a , . . . , a n in unary, Unary BinPacking asks the existence of a partition S , . . . , S t of { , , . . . , n } such that (cid:80) i ∈ S j a i = t (cid:80) ≤ i ≤ n a i for 1 ≤ j ≤ t . Unary Bin Packing is W[1]-hardparameterized by t [37].We assume that t ≥ a i are given in unary. Let B = t (cid:80) ≤ i ≤ n a i and W = ( t − B = (cid:80) ≤ i ≤ n a i − B . The assumption t ≥ B ≤ W/ a i ≥ B for some i , then the instance is a trivial no instance(when a i > B ) or the element a i is irrelevant (when a i = B ). Hence, we assumethat a i < B (and thus a i < W/
2) for every i .The reduction to Unary MMOO is depicted in Fig. 3. From the integers a , a , . . . , a n , we construct the graph obtained from a complete bipartite graphon the vertex set { u, s , s , . . . , s t } ∪ { v , . . . , v n } by adding the edge { u, s } . Weset w ( { v i , s j } ) = a i for all i, j , w ( { v i , u } ) = W − a i for all i , and w ( { u, s } ) = W .The vertices s , s , . . . , s t , u form a vertex cover of size t + 1. We set the targetmaximum outdegree r to W . We show that this instance of Unary MMOO isa yes instance if and only if there exists a partition S , . . . , S t of { , , . . . , n } such that (cid:80) i ∈ S j a i = B for all j . Intuitively, we can translate the solutions of In [50], W[1]-hardness was stated for tw but the proof shows it for td as well.0 Gima et al. s v v v n ua a a n ... W − a W − a W − a n ... s s t W Fig. 3.
Reduction from
Unary Bin Packing to Unary MMOO . the problems by picking a i into S j if { v i , s j } is oriented from v i to s j , and viceversa.Assume that there exists a partition S , . . . , S t of { , , . . . , n } such that (cid:80) i ∈ S j a i = B for all j . We first orient the edge { u, s } from u to s and eachedge { v i , u } from v i to u . (See the thick edges in Fig. 3.) Then, we orient { v i , s j } from v i to s j if and only if i ∈ S j . Under this orientation, all vertices haveoutdegree exactly W : a i + ( W − a i ) for each v i and (cid:80) i/ ∈ S j a i = (cid:80) ≤ i ≤ n a i − B for each s j .Conversely, assume that there is an orientation such that each vertex hasoutdegree at most W . Since the sum of the edge weights is ( n + t + 1) W andthe graph has n + t + 1 vertices, the outdegree of each vertex has to be exactly W . Since a i < W/ i , each edge { v i , u } has weight larger than W/ u , the only way to obtain outdegree exactly W is to orient { u, s } from u to s and { v i , u } from v i to u for all i . Furthermore, for each i , thereexists exactly one vertex s j such that { v i , s j } is oriented from v i to s j . Let S j ⊆ { , , . . . , n } be the set of indices i such that { v i , s j } is oriented from v i to s j . The discussion above implies that S , . . . , S t is a partition of { , . . . , n } .The outdegree of s j is (cid:80) i/ ∈ S j a i , which is equal to W = (cid:80) ≤ i ≤ n a i − B . Thus, (cid:80) i ∈ S j a i = (cid:80) ≤ i ≤ n a i − W = B . (cid:117)(cid:116) Let G = ( V, E ) be a graph. Given a linear ordering σ on V , the stretch of { u, v } ∈ E , denoted str σ ( { u, v } ), is | σ ( u ) − σ ( v ) | . The bandwidth of G , denoted bw ( G ), is defined as min σ max e ∈ E str σ ( e ), where the minimum is taken over alllinear orderings on V . Given a graph G and an integer w , Bandwidth askswhether bw ( G ) ≤ w . Bandwidth is NP-complete on trees of pw = 3 [45] andon graphs of pw = 2 [46]. Fellows et al. [27] presented an FPT algorithm for Bandwidth parameterized by vc . Here we show that Bandwidth is W[1]-hardparameterized by td on trees. The proof is inspired by the one by Muradian [46]. Theorem 5.1.
Bandwidth is W[1]-hard parameterized by td on trees. reedepth, Vertex Cover, and Vertex Integrity 11 Proof.
Let ( a , . . . , a n ; t ) be an instance of Unary Bin Packing with t ≥ B = t (cid:80) ≤ i ≤ n a i be the target weight. We construct an equivalent instance( T = ( V, E ) , w ) of Bandwidth as follows (see Fig. 4). We start with a path( z , x , y , z , . . . , x t , y t , z t ) of length 3 t . For 1 ≤ i ≤ t −
1, we attach 12 tnB leaves to z i . To z and z t , we attach 12 tnB + 4 n + 1 leaves. For 1 ≤ i ≤ n ,we take a star with 6 tn · a i − v i . Finally, we connect each v i to x with a path with 6 t − w to6 tnB + 2 n + 1. Note that | V | = (3 t + 2) w + 1.We can see an upper bound of td ( T ) as follows. We remove x and all theleaves from T . This decreases treedepth by at most 2. The remaining graph isa disjoint union of paths and a longest path has order 6 t −
3. Since td ( P n ) = (cid:100) log ( n + 1) (cid:101) [47], we have td ( T ) ≤ (cid:100) log (6 t − (cid:101) ≤ log t + 6.Now we show that ( T, w ) is a yes instance of of
Bandwidth if and only if( a , . . . , a n ; t ) is a yes instance of Unary Bin Packing .( = ⇒ ) First assume that bw ( T ) ≤ w and that σ is a linear ordering on V such that max e ∈ E str σ ( e ) ≤ w . Since deg( z ) = 12 tnB + 4 n + 2 = 2 w , its closedneighborhood N [ z ] has to appear in σ consecutively, where z appears at themiddle of this subordering. Furthermore, no edge can connect a vertex appearingbefore z in σ and a vertex appearing after z as such an edge has stretchlarger than w . Since the edges not incident to z form a connected subgraph,we can conclude that the vertices in V − N [ z ] appear either all before N [ z ]or all after N [ z ] in σ . By symmetry, we can assume that those vertices appearafter N [ z ] in σ . This implies that σ ( z ) = w + 1. By the same argument,we can show that all vertices in N [ z t ] appear consecutively in the end of σ and σ ( z t ) = | V | − w = (3 t + 1) w + 1. Since σ ( z t ) − σ ( z ) = 3 tw and thepath ( z , x , y , z , . . . , x t , y t , z t ) has length 3 t , each edge in this path has stretchexactly w in σ . Namely, σ ( x i ) = (3 i − w + 1, σ ( y i ) = 3 iw + 1, and σ ( z i ) =(3 i + 1) w + 1.For each leaf (cid:96) attached to z i (1 ≤ i ≤ t − σ ( y i ) < σ ( (cid:96) ) < σ ( x i +1 ) holds.Other than these leaves, there are 2( w − − tnB = 4 n vertices placed between y i and x i +1 . Let V i be the set consisting of v i and the leaves attached to it. For j ∈ { , . . . , t } , let I j be the set of indices i such that v i is put between z j − and · · · tn · a − · · · tn · a n − · · · · · · tnB · · · tnB · · · tnB · · · tnB + 4 n + 1 · · · tnB + 4 n + 1 · · · t − (cid:124)(cid:123)(cid:122)(cid:125) x y z x y z x t y t z t − z t z target width w = 6 tnB + 2 n + 1 v v n V V n Fig. 4.
Reductions from
Unary Bin Packing to Bandwidth .2 Gima et al. x y z x y z x y v i Fig. 5.
Embedding the path from x to v i . The gray boxes are the occupied positionand the white points are the vacant positions. ( n = 2, j = 2, t = 3.) z j . If i ∈ I j , then all 6 tn · a i vertices in V i are put between y j − and x j +1 . (Weset y := z .)If (cid:80) i ∈ I j a i ≥ B + 1, then | (cid:83) i ∈ I j V i | ≥ tn ( B + 1) > w + 8 n − t ≥ y j − and x j +1 after putting theleaves attached to z j − and z j : we can put at most 4 n vertices between y j − and x j , at most 4 n vertices between y j and x j +1 , and at most w − x j and y j . Since I , . . . , I t form a partition of { , . . . , n } and (cid:80) ≤ i ≤ n a i = tB ,we can conclude that (cid:80) i ∈ I j a i = B for 1 ≤ j ≤ t .( ⇐ = ) Next assume that there exists a partition S , . . . , S t of { , , . . . , n } such that (cid:80) i ∈ S j a i = B for all 1 ≤ j ≤ t .We put N [ z ] at the beginning of σ and N [ z t ] at the end. We set σ ( x i ) =(3 i − w + 1, σ ( y i ) = 3 iw + 1, and σ ( z i ) = (3 i + 1) w + 1. For 1 ≤ i ≤ t −
1, weput the leaves attached to z i so that a half of them have the first 6 tnB positionsbetween y i and z i and the other half have the first 6 tnB positions between z i and x i +1 . For each S j , we put the vertices in (cid:83) i ∈ S j V i so that they take the first6 tnB positions between x j and y j .Now we have 2 n vacant positions at the end of each interval between x i and y i for 1 ≤ i ≤ t , between y i and z i for 1 ≤ i ≤ t −
1, and between z i and x i +1 for1 ≤ i ≤ t −
1. To these positions, we need to put the inner vertices of the pathsconnecting x and v , . . . , v n . Let P i be the inner part of x – v i path. The path P i uses the (2 i − i )th vacant positions in each interval as follows (seeFig. 5).Let i ∈ S j . Starting from x , P i proceeds from left to right and visits thetwo positions in each interval consecutively until it arrives the interval between x j and y j . At the interval between x j and y j , P i switches to the phase whereit only visits the (2 i )th vacant position in each interval and still proceeds fromleft to right until it reaches the interval between x t and y t . Then P i changes thedirection and switches to the phase where it visits the (2 i − x j and y j .Now all the vertices are put at distinct positions and it is easy to see that noedge has stretch more than w . This completes the proof. (cid:117)(cid:116) Using vertex integrity as a structural graph parameter, we presented finer anal-yses of the parameterized complexity of well-studied problems. Although weneeded a case-by-case analysis depending on individual problems, the results in reedepth, Vertex Cover, and Vertex Integrity 13 this paper would be useful for obtaining a general method to deal with vertexintegrity.Although we succeeded to extend many fixed-parameter algorithms param-eterized by vc to the ones parameterized by vi , we were not so successful ongraph layout problems. Fellows et al. [27] showed that Imbalance , Bandwidth , Cutwidth , and
Distortion are fixed-parameter tractable parameterized by vc . Lokshtanov [41] showed that Optimal Linear Arrangement is fixed-parameter tractable parameterized by vc . Are these problems fixed-parametertractable parameterized by vi ? We answered only for Imbalance in this paper.
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We give formal definitions of vc ( G ), td ( G ), and vi ( G ) only. See [36] for thedefinitions of pw ( G ), tw ( G ), and clique-width cw ( G ). A.1 Vertex cover
Let G = ( V, E ) be a graph. A set S ⊆ V is a vertex cover of G if each componentof G − S has exactly one vertex. The vertex cover number of G , denoted vc ( G ),is the size of a minimum vertex cover of G . It is known that a vertex cover ofsize k (if exists) can be found in time O (2 k · n ) [19], where n = | V | (see [15]for the currently fastest algorithm). Thus we can assume that a vertex cover ofminimum size is given when designing an algorithm parameterized by vc . A.2 Treedepth
The treedepth of a graph G = ( V, E ), denoted td ( G ), is defined recursively asfollows: td ( G ) = | V | = 1 , max ≤ i ≤ c td ( C i ) G has c ≥ C , . . . , C c , v ∈ V td ( G − v ) otherwise . In other words, a graph G = ( V, E ) has treedepth at most d if there is a rootedforest F of height at most d on the same vertex set V such that two verticesare adjacent in G only if one is an ancestor of the other in F . It is known thatsuch F , if exists, can be found in time 2 O ( d ) · n [49], where n = | V | . So weassume that such a rooted forest of depth td ( G ) is given together with G whenthe parameter is td .From the rooted forest F , one can easily construct a path decomposition of G with maximum bag size at most d : use leaves as bags and put all ancestors of aleaf into the bag corresponding to the leaf. This implies that pw ( G ) + 1 ≤ td ( G )for every graph G . On the other hand, td cannot be bounded by any function of reedepth, Vertex Cover, and Vertex Integrity 17 pw in general. For example, pw ( P n ) = 1 and td ( P n ) = (cid:100) log ( n + 1) (cid:101) [47], where P n is the path of order n .In general, we have the following upper bound of the length of paths. Proposition A.1 ([47]).
The length of a longest path in G is less than td ( G ) . A.3 Vertex integrity
The vertex integrity [5] of a graph G , denoted vi ( G ), is the minimum integer k satisfying that there is a vertex set S ⊆ V ( G ) such that | S | + | V ( C ) | ≤ k foreach component C of G − S . We call such S a vi ( k ) -set of G . For an n -vertexgraph of vertex integrity at most k , we can find a vi ( k ) -set in O ( k k +1 n ) time [22].Hence, without loss of generality, we can assume that a vi ( k )-set is given as apart of input when designing an FPT algorithm parameterized by vi . Observethat td ( G ) ≤ vi ( G ) since we can first remove the k (cid:48) ≤ k vertices in a vi ( k )-setand then each component has order at most k − k (cid:48) and thus treedepth at most k − k (cid:48) . Also, since a vertex cover of size k is a vi ( k + 1)-set, vi ( G ) ≤ vc ( G ) + 1holds.Dvoˇr´ak et al. [23] showed that Integer Linear Programming (ILP) isfixed-parameter tractable parameterized by the fracture number of the incidence,which is basically equivalent to the vertex integrity. Ganian et al. [33] showedthat
Bounded Degree Deletion is W[1]-hard parameterized by treedepthbut fixed-parameter tractable parameterized by core fracture number, whichcan be seen as a generalization of vertex integrity. Bodlaender et al. [12] showedthat
Subgraph Isomorphism is fixed-parameter tractable parameterized by thevertex integrity of both graphs, while the problem is NP-complete for graphs oftreedepth 3.
B ILP parameterized by the number of variables
Lenstra [40] showed that the feasibility of an integer linear programming (ILP)formula can be decided in FPT time when parameterized by the number ofvariables. The time and space complexity was later improved by Kannan [38]and by Frank and Tardos [29]. Their algorithms can be used also for the followingILP optimization problem (see e.g., [27]). p -Opt-ILP Input:
A matrix A ∈ Z m × p , vectors b ∈ Z m and c ∈ Z p . Task:
Find a vector x ∈ Z p that minimizes c (cid:62) x and satisfies that Ax ≥ b . Proposition B.1 ([40,38,29]). p -Opt-ILP can be solved using O ( p . p + o ( p ) · L · log( M N )) arithmetic operations and space polynomial in L , where L is thenumber of bits in the input, N is the maximum absolute value any variable cantake, and M is an upper bound on the absolute value of the minimum taken bythe objective function. C Omitted proofs in Section 3
Theorem C.1.
Maximum Common Induced Subgraph is fixed-parametertractable parameterized by the sum of the vertex integrity of input graphs.Proof.
Since the proof is almost the same with the one for Theorem 3.1, here weonly describe the differences for handling induced subgraphs.Let G = ( V , E ) and G = ( V , E ) be the input graphs of vertex integrityat most k . We will find U ⊆ V and U ⊆ V with maximum size | U | = | U | such that there is an isomorphism η from G [ U ] to G [ U ]. Step 1. Guessing matched vi (2 k ) -sets R and R . In the same way as before,we guess vi (2 k )-sets R and R of G and G , respectively, and a bijection η | R : R → R . The only difference here is that we reject the current guess if η | R is not an isomorphism from G [ R ] to G [ R ]. Step 2. Extending the guessed parts of η . To handle induced subgraphs, we needto modify the definition of “to decompose” as follows: for a type- t component C of H i − R i and a multiset T of g-( H i , R i )-types, we say that η : U → U decomposes C into T if T coincides with the multiset of g-( H i , R i )-types of thepairs ( A, B ) such that A is a component of H i [ V ( C ) ∩ U i ] and B is the set of alledges connecting A and R i . Everything else works as before. (cid:117)(cid:116) D Omitted proofs in Section 4
Theorem D.1.
Binary MMOO is NP-complete for graphs of vc = 3 .Proof. Since the problem clearly belongs to NP, we show the NP-hardness bypresenting a reduction from
Partition , which is NP-complete [34]. Given aneven number of positive integers a , a , . . . , a n in binary, Partition asks the ex-istence of a partition { S , S } of { , , . . . , n } such that (cid:80) i ∈ S j a i = (cid:80) ≤ i ≤ n a i for j ∈ { , } . This problem remains NP-hard with an additional condition | S | = | S | = n/ n ≥
10 since otherwise the problem canbe solved in polynomial time. Let B = W = (cid:80) ≤ i ≤ n a i .The proof is almost the same with the one of Theorem 4.1 except that t = 2.Observe that we assumed there that t ≥ a i < W/ i . To have this assumption here, we start with an instance a , a , . . . , a n of Partition with the restriction | S | = | S | = n/
2. Let a (cid:48) i = a i + B for each i , and B (cid:48) = W (cid:48) = (cid:80) ≤ i ≤ n a (cid:48) i = ( n/ B . Clearly, this is an equivalent instance aswe added the same value to each number. Also, a (cid:48) i < W (cid:48) / i since n ≥
10 and a i < B imply that a (cid:48) i = a i + B < B ≤ ( n/ B/ W (cid:48) /
2. Nowwe observe that the restriction | S | = | S | = n/ (cid:80) i ∈ S a (cid:48) i = B (cid:48) for some S ⊆ { , . . . , n } , then | S | = n/ S (cid:54) = n/
2. By swapping S and { , . . . , n } \ S if necessary, wecan assume that S ≤ n/ −
1. This gives ( n/ B = (cid:80) i ∈ S a (cid:48) i ≤ ( n/ − B + (cid:80) i ∈ S a i , which implies (cid:80) i ∈ S a i ≥ B = (cid:80) ≤ i ≤ n a i , a contradiction. reedepth, Vertex Cover, and Vertex Integrity 19 We construct an instance of
Binary MMOO as exactly we did in the proofof Theorem 4.1 by setting t = 2 and using a (cid:48) i , B (cid:48) , and W (cid:48) instead of a i , B , and W . The equivalence of the instances can be shown in the same way. (cid:117)(cid:116) Theorem D.2.
Binary MMOO can be solved in polynomial time for graphsof vc ≤ .Proof. Let G = ( V, E ), w : E → Z + , r ∈ Z + be an instance of Binary MMOO .We assume that w ( e ) ≤ r for each e ∈ E since otherwise the problem is trivial.If there is a vertex of degree at most 1, we can safely remove it from the graphsince we can always orient the edge incident to the vertex (if exists) from thevertex to the other endpoint. Hence, we assume that G has minimum degree atleast 2.Let { p, q } ⊆ V be a vertex cover of G . By the assumption on the minimumdegree, every vertex v ∈ V \ { p, q } is adjacent to both p and q . If w ( { v, p } ) + w ( { v, q } ) ≤ r , then we can safely orient the edges from v to p and q . Thus weremove such vertices from the graph. Now it holds that w ( { v, p } ) + w ( { v, q } ) > r for all v ∈ V \ { p, q } . In particular, max { w ( { v, p } ) , w ( { v, q } ) } > r/ v ∈ V \ { p, q } .Observe that for each vertex, at most one edge of weight more than r/ p and q , we guess suchedges. That is, we guess one edge of weight more than r/ p ( q ,resp.) and orient it from p ( q , resp.) to the other endpoint; or guess that thereis no such edge. These guesses determine almost a complete orientation. For anon-guessed edge { v, p } with w ( { v, p } ) > r/
2, we orient it from v to p . Since w ( { v, p } ) + w ( { v, q } ) > r , we then have to orient { v, q } from q to v . The othercase of w ( { v, q } ) > r/ { p, q } (if they are adjacent). We just try both directions of { p, q } and check if the whole orientation is of maximum outdegree at most r . (cid:117)(cid:116) E Extending algorithms known for vc parameterizations E.1 Capacitated problems
Let G = ( V, E ) be a graph with a capacity function c : V → Z + such that c ( v ) ≤ deg( v ) for each v ∈ V . A set C ⊆ V is a capacitated vertex cover ifthere exists a mapping f : E → C such that f ( e ) is an endpoint of e for each e ∈ E and |{ e ∈ E | f ( e ) = v }| ≤ c ( v ) for each v ∈ C . A set D ⊆ V is a capacitated dominating set if there exists a mapping f : V \ D → D such that f ( v ) ∈ N ( v ) ∩ D for each v ∈ V \ D and |{ v ∈ V \ D | f ( v ) = u }| ≤ c ( u ) foreach u ∈ D . Now the problems studied in this section are defined as follows. Capacitated Vertex Cover
Input:
A graph G , a capacity function c : V → Z + , a positive integer k . Question:
Is there a capacitated vertex cover X of G with | X | ≤ k ? Capacitated Dominating Set
Input:
A graph G , a capacity function c : V → Z + , a positive integer k . Question:
Is there a capacitated dominating set D of G with | D | ≤ k ?It is known that Capacitated Vertex Cover is W[1]-hard parameterizedby td , and Capacitated Dominating Set is W[1]-hard parameterized by td + k [20]. For a vertex set S of G , we say that components C and C of G − S havethe same c -type if C and C have the same ( G, S )-type and furthermore thereis an isomorphism g from G [ S ∪ V ( C )] to G [ S ∪ V ( C )] such that g | S is theidentity and c ( v ) = c ( g ( v )) for each v ∈ S ∪ V ( C ). We say that a component C of G − S is of c -type t by using a canonical form t of the members of the c -typeequivalence class of C . If S is a vi ( k )-set of G , then every vertex in G − S hasdegree less than k in G , and thus its capacity is also less than k . This impliesthat the number of different c -types depends only on k . Theorem E.1.
Capacitated Vertex Cover is fixed-parameter tractable pa-rameterized by vi .Proof. We are going to find a minimum capacitated vertex cover X of G . Let S bea vi ( k )-set of the input graph G = ( V, E ). We first guess the subset X S = X ∩ S and the partial mapping f S : E ( G [ S ]) → X S with f S ( e ) ∈ e for each e ∈ E ( G [ S ]).The numbers of candidates for X S and f S depend only on k . For each v ∈ X S ,we set c (cid:48) ( v ) = c ( v ) − |{ e ∈ E ( G [ S ]) | f S ( e ) = v }| . Each v ∈ X S can cover c (cid:48) ( v )edges between S and V − S .Let C be a c -type t component of G − S . We say that a pair ( W, f ) of asubset W ⊆ V ( C ) and a mapping f : E ( C ) ∪ E ( V ( C ) , S ) → W ∪ X S is feasible if |{ e | f ( e ) = v }| ≤ c ( v ) for each v ∈ W and f ( e ) ∈ e for each e ∈ E ( C ) ∪ E ( C, S ). The number of feasible pairs depends only on k . A feasible pair gives a cover ofall edges in C and some edges between V ( C ) and S , and it asks X S to coverthe remaining edges between V ( C ) and S in a certain way. Now it suffices tofind an assignment of feasible pairs to components of G − S that minimizes thenumber of vertices used by the feasible pairs and does not exceed the capacityof any vertex in X S .We represent by a nonnegative variable x t,W,f the number of c -type t com-ponents C of G − S such that V ( C ) ∩ X = W and ( W, f ) is a feasible pair. Thenumber of such variables depends only on k . Let d t be the number of componentsof c -type t in G − S . Since each component of G − S has to be assigned a feasiblepair, we have the following constraints: (cid:80) W, f x t,W,f = d t for each c -type t. The capacity constraints for X S can be expressed as follows: (cid:80) t, W, f f, v ) · x t,W,f ≤ c (cid:48) ( v ) for each v ∈ X S , The W[1]-hardness results are stated only for tw and tw + k but the proofs actuallyshow them for td and td + k , respectively. For vertex sets A and B , E ( A, B ) denotes the set of edges between A and B .reedepth, Vertex Cover, and Vertex Integrity 21 where f, v ) is the number of edges that f maps to v . Finally, our objectivefunction to minimize is | X S | + (cid:80) t, W, f | W | · x t,W,f .By finding an optimal solution to the ILP above for each guess of X S and f S , we can find the minimum capacitated vertex cover of G . Since the numberof guesses and the number of variables depend only on k , the theorem followsby Proposition B.1. (cid:117)(cid:116) Theorem E.2.
Capacitated Dominating Set is fixed-parameter tractableparameterized by vi Proof.
We are going to find a minimum capacitated dominating set D of G . Let S be a vi ( k )-set of the input graph G = ( V, E ).We first guess the partition ( D S , A S , B S ) of S such that D S = D ∩ S , A S isthe set of vertices dominated by D S , and B S is the set of vertices dominated by D \ S . Next we guess the partial mapping f S : A S → D S with f S ( v ) ∈ N ( v ) ∩ D S for each v ∈ A S . The numbers of candidates for ( D S , A S , B S ) and f S dependonly on k . For each v ∈ D S , we set c (cid:48) ( v ) = c ( v ) − |{ u ∈ A S | f S ( u ) = v }| . Each v ∈ D S can dominate c (cid:48) ( v ) vertices in V − S .Let C be a c -type t component of G − S . Let ( D C , A C , B C ) be a parti-tion of V ( C ), B (cid:48) S ⊆ B S , f : A C ∪ B (cid:48) S → D C , and g : B C → D S . We say that( D C , A C , B C , B (cid:48) S , f, g ) is feasible if f ( v ) ∈ N ( v ) ∩ D C for each v ∈ A C ∪ B (cid:48) S , g ( v ) ∈ N ( v ) ∩ D S for each v ∈ B C , and |{ u ∈ A C ∪ B (cid:48) S | f ( u ) = v }| ≤ c ( v )for each v ∈ D C . The number of feasible tuples depends only on k . A feasibletuple gives a domination of all vertices in V ( C ) \ B C and B (cid:48) S , and it asks D S todominate B C in a certain way.As before, it suffices to find an assignment of feasible tuples to componentsof G − S that minimizes the number of vertices used by the feasible tuples anddoes not exceed the capacity of any vertex in D S .We represent by a nonnegative variable x t,D C ,A C ,B C ,B (cid:48) S ,f,g the number of c -type t components C of G − S such that V ( C ) ∩ D = D C and ( D C , A C , B C , B (cid:48) S , f, g )is a feasible tuple. The number of such variables depends only on k . Let d t bethe number of components of c -type t in G − S . Since each component of G − S has to be assigned a feasible tuple, we have the following constraints: (cid:88) D C , A C , B C , B (cid:48) S , f, g x t,D C ,A C ,B C ,B (cid:48) S ,f,g = d t for each c -type t. The capacity constraints for D S can be expressed as follows: (cid:88) D C , A C , B C , B (cid:48) S , f, g g, v ) · x t,D C ,A C ,B C ,B (cid:48) S ,f,g ≤ c (cid:48) ( v ) for each v ∈ D S , where g, v ) is the number of vertices that g maps to v . We also have toguarantee that each vertex in B S is dominated by a vertex in V − S . This canbe done by the following constraints: (cid:88) D C , A C , B C , B (cid:48) S (cid:51) v, f, g x t,D C ,A C ,B C ,B (cid:48) S ,f,g ≥ v ∈ B S . Finally, our objective function to minimize is | D S | + (cid:88) D C , A C , B C , B (cid:48) S , f, g | D C | · x t,D C ,A C ,B C ,B (cid:48) S ,f,g . As before the discussion so far implies the theorem. (cid:117)(cid:116)
E.2 Coloring and partitioning problems
Precoloring Extension , Equitable Coloring , and
Equitable ConnectedPartition form a first set of problems studied under the “treewidth versusvertex cover” perspective [25,26,28].
Equitable Coloring and
Precolor-ing Extension are fixed-parameter tractable parameterized by vc [28] andW[1]-hard parameterized by td [26]. Equitable Connected Partition isfixed-parameter tractable parameterized by vc and W[1]-hard parameterized by pw [25].In this section, we show that all the three problems are fixed-parametertractable parameterized by vi . Precoloring Extension
Given a graph G = ( V, E ), a precoloring c U : U →{ , . . . , r } for some U ⊆ V , and a positive integer r , Precoloring Extension asks whether G admits a proper r -coloring c such that c ( v ) = c U ( v ) for every v ∈ U . Theorem E.3.
Precoloring Extension is fixed-parameter tractable param-eterized by vi .Proof. Let ( G = ( V, E ) , c U , r ) be an instance of Precoloring Extension . Let S be a vi ( k )-set of G . For each v ∈ V , let L ( v ) be the following set (the list ofallowed colors): L ( v ) = { c U ( v ) } v ∈ U, { , . . . , r } \ { c U ( u ) | u ∈ N ( v ) ∩ U } v ∈ S \ U, { , . . . , min { r, k }} \ { c U ( u ) | u ∈ N ( v ) ∩ U } v ∈ V \ ( S ∪ U ) . Observe that there exists a proper r -coloring c of G with c ( v ) = c U ( v ) for all v ∈ U if and only if there is a proper coloring c (cid:48) of G with c (cid:48) ( v ) ∈ L ( v ). Thisis almost trivial except for the case of v ∈ V \ ( S ∪ U ), where we restrict thedomain to { , . . . , k } when k < r . This can be justified by considering the degreeof v . Since S is a vi ( k )-set and v / ∈ S , we have deg( v ) < k . Thus, after coloring G − v , v can always be colored with a color not used in its neighborhood.Now in the list coloring setting, we can remove the vertices in U unless theinstance is a trivial no instance with { u, v } ∈ E such that c U ( u ) = c U ( v ). In The W[1]-hardness results are stated only for tw but the proofs actually show themfor td .reedepth, Vertex Cover, and Vertex Integrity 23 the following, we consider the graph where U is removed and still use the samesymbols G and S .Let v be a vertex with | L ( v ) | ≥ k . By the definition of L , v ∈ S . Such avertex can be safely removed: the vertices in V \ S use colors only in { , . . . , k } ;and the vertices in S − v use at most k − L ( v ). We now assume that L ( u ) < k for all vertices in the graph.Now we guess the coloring of S and then check independently for each com-ponent C of G − S whether G [ S ∪ V ( C )] has a coloring consistent with L and theguessed coloring of S . The number of possible colorings of S is at most (2 k ) k .Since | S ∪ V ( C ) | ≤ k , checking the existence of a consistent coloring can be donein time depending only on k . (cid:117)(cid:116) Equitable Coloring
Given an n -vertex graph G = ( V, E ) and a positive integer r , Equitable Coloring asks whether G admits a proper r -coloring c such that |{ v ∈ V | c ( v ) = i }| ∈ {(cid:98) n/r (cid:99) , (cid:100) n/r (cid:101)} for each i ∈ { , . . . , r } . We call such acoloring an equitable r -coloring . Theorem E.4.
Equitable Coloring is fixed-parameter tractable parameter-ized by vi .Proof. Let ( G = ( V, E ) , r ) be an instance of Equitable Coloring , and S be a vi ( k )-set of G . We split the proof into two cases: r ≤ k and r > k . We reduceboth cases to the feasibility test of the ILP defined as follows. By Proposition ?? ,the theorem will follow. Case 1: r ≤ k . We guess a partition S , . . . , S r of S such that each of them isan independent set and some of them may be empty. Since | S | ≤ k and r ≤ k ,the number of such partitions depends only on k . We interpret this partition asa coloring of G [ S ] and try to extend this to the whole graph.For a ( G, S )-type t , a coloring µ : V ( C ) → { , . . . , r } of a type- t component C of G − S is feasible if S i ∪ { v ∈ V ( C ) | µ ( v ) = i } is an independent set foreach i . We set µ i = |{ v ∈ V ( C ) | µ ( v ) = i }| .We represent by a nonnegative variable x t,µ the number of type- t componentscolored with a feasible µ . Since each component of G − S has to be colored, wehave the following constraints: (cid:80) µ x t,µ = d t for each ( G, S )-type t, where d t is the number of type- t components in G − S . The equitable constraintscan be expressed as follows: (cid:80) t,µ µ i · x t,µ = (cid:100) n/r (cid:101) − | S i | for each i ∈ { , . . . , b } , (cid:80) t,µ µ i · x t,µ = (cid:98) n/r (cid:99) − | S i | for each i ∈ { b + 1 , . . . , r } , where b is the remainder of n/r . Case 2: r > k . In this case, we do not have an upper bound of r . The first trickis that we can still guess the coloring of S since we use at most k colors there.The second trick is that after checking the extendability of the k colors, the restof the problem becomes trivial.We guess a partition S , . . . , S k of S and an integer a such that: each S i is a possibly-empty independent set; and there are disjoint independent sets W , . . . , W k such that S i ⊆ W i for all i , | W i | = (cid:100) n/r (cid:101) for 1 ≤ i ≤ a , and | W i | = (cid:98) n/r (cid:99) for a + 1 ≤ i ≤ k . Then, G (cid:48) := G − (cid:83) ≤ i ≤ k W i has an equitable r − k coloring if and only if G has an equitable r coloring having W , . . . , W k ascolor classes.We can show that actually G (cid:48) always has an equitable r − k coloring. Observethat each component in G (cid:48) has order at most k < r − k as G (cid:48) is a subgraph of G − S . We now linearly order the vertices of G (cid:48) in such a way that the vertices ofa component appear consecutively. Then we color the first vertex in this orderingwith color 1, the second one with color 2, and so on. Formally, we color the i thvertex in this ordering with color ( i mod ( r − k )) + 1. Since each component hasorder less than r − k , we never repeat a color in a component. Thus, this is anequitable r − k coloring of G (cid:48) . Therefore, it suffices to decide whether there existsthe super sets W , . . . , W k .Since we are searching for a partial coloring, we use a special character ∗ to indicate “not colored.” We need to change the definition of feasibility. For a( G, S )-type t , a coloring µ : V ( C ) → {∗} ∪ { , . . . , k } of a type- t component C of G − S is feasible , if S i ∪ { v ∈ V ( C ) | µ ( v ) = i } is an independent set for each i (cid:54) = ∗ . We set µ i = |{ v ∈ V ( C ) | µ ( v ) = i }| . Now the rest of the proof is exactlythe same as before.We represent by a nonnegative variable x t,µ the number of type- t componentscolored with a feasible µ . Since each component of G − S has to be colored,we have the following constraints: (cid:80) µ x t,µ = d t for each ( G, S )-type t , where d t is the number of type- t components in G − S . The equitable constraintscan be expressed as follows: (cid:80) t,µ µ i · x t,µ = (cid:100) n/r (cid:101) − | S i | for 1 ≤ i ≤ a , and (cid:80) t,µ µ i · x t,µ = (cid:98) n/r (cid:99) − | S i | for a + 1 ≤ i ≤ k . (cid:117)(cid:116) Equitable Connected Partition
Given an n -vertex graph G = ( V, E ) and apositive integer r , Equitable Connected Partition asks whether there is apartition of V , . . . , V r of V such that G [ V i ] is connected and | V i | ∈ {(cid:98) n/r (cid:99) , (cid:100) n/r (cid:101)} for all i . We call such a partition an equitable connected r -partition . Theorem E.5.
Equitable Connected Partition is fixed-parameter tractableparameterized by vi .Proof. Let ( G = ( V, E ) , r ) be an instance of Equitable Connected Par-tition , and S be a vi ( k )-set of G . Observe that at most k of V , . . . , V r canintersect S . We split the proof into two cases r ≤ k and r > k . Case 1: r ≤ k . If additionally (cid:98) n/r (cid:99) ≤ k holds in this case, then n ∈ O ( k ).Thus we assume that (cid:98) n/r (cid:99) > k . This implies that every V i intersects S . Wefirst guess the partition S , . . . , S r of S . reedepth, Vertex Cover, and Vertex Integrity 25 Let C and C be components of G − S with the same type, and µ j : V ( C j ) →{ , . . . , r } for each j ∈ { , } . Then, we say that ( C , µ ) and ( C , µ ) are equiv-alent if there is an isomorphism η from G [ S ∪ C ] and G [ S ∪ C ] such that η fixes S and µ ( v ) = µ ( η ( v )) for all v ∈ V ( C ). A set M = { ( C , µ ) , . . . , ( C p , µ p ) } is feasible if C j is a component of G − S for each j , µ j : V ( C j ) → { , . . . , r } foreach j , and the subgraph of G induced by S i ∪ (cid:83) ≤ j ≤ p { v ∈ V ( C j ) | µ j ( v ) = i } is connected for all 1 ≤ i ≤ r . Let M (cid:48) = { ( C (cid:48) , µ (cid:48) ) , . . . , ( C (cid:48) q , µ (cid:48) q ) } be a subset of M obtained by removing all but one of each equivalent class. It is easy to seethat M (cid:48) is feasible if and only if so is M . Let t (cid:48) j be the ( G, S )-type of C (cid:48) j . Wecall the set { ( t (cid:48) , µ (cid:48) ) , . . . , ( t (cid:48) q , µ (cid:48) q ) } a type-color representation of M .Now we guess the type-color representation T = { ( t , µ ) , . . . , ( t q , µ q ) } of asolution. That is, we find a partition such that at least one component of type t is partitioned by µ , and no component is partitioned in a way not includedin T . The number of candidates depends only on k , and the feasibility of eachcandidate can be checked in polynomial time.By a nonnegative variable x t,µ for ( t, µ ) ∈ T , we represent the number oftype- t components that we partition by µ or an equivalent mapping. Since wetake at least one such partition of type- t components, we set the constraint x t,µ ≥ t, µ ) ∈ T . Now the connectivity has been handled, and we onlyneed to force the equitable partition. Since each component has to be partitioned,we need the following constraints: (cid:80) ( t,µ ) ∈T x t,µ = d t for each type t, where d t is the number of type- t components in G − S . The equitable constraintscan be expressed as follows: (cid:80) ( t,µ ) ∈T µ ( i ) · x t,µ = (cid:100) n/r (cid:101) − | S i | for each i ∈ { , . . . , a } , (cid:80) ( t,µ ) ∈T µ ( i ) · x t,µ = (cid:98) n/r (cid:99) − | S i | for each i ∈ { a + 1 , . . . , r } , where a is the remainder of n/r and µ ( i ) is the number of vertices µ maps to i .Since the number of variables depends only on k , Proposition ?? impliesthat the feasibility test of the ILP defined above is fixed-parameter tractableparameterized by k . Case 2: r > k . In this case, some V i does not intersect S , and thus it is containedin a component of G − S . This implies that (cid:98) n/r (cid:99) ≤ k , and thus max i | V i | ≤ k +1.We first guess the number k (cid:48) < k of the V i ’s intersecting S and the number a ≤ k (cid:48) of size (cid:100) n/r (cid:101) sets among them. Now we guess V : guess at most k + 1 types of G − S ; guess the number of components we take from the chosen types, which isat most k + 1; and for each component, guess the subset of the vertices taken to V . The number of candidates depends only on k . In general, when we guess V i ,2 ≤ i ≤ k (cid:48) , we first remove the vertices chosen for (cid:83) ≤ j ≤ i − V j and recomputeand redefine the types. Then, we can guess V i in exactly the same way as thecase of i = 1. The number of candidates for all V , . . . , V k (cid:48) depends only on k .We reject the guess if some G [ V i ] is disconnected. Let W = (cid:83) ≤ j ≤ k (cid:48) V j . Now it suffices to decide whether G − W has an equi-table connected ( r − k (cid:48) )-partition. For each component C of G − W , we enumerateall the possible pairs ( p, q ) of nonnegative integers such that C admits an equi-table connected ( p + q )-partition such that p parts have size (cid:100) n/r (cid:101) and q partshave size (cid:98) n/r (cid:99) . This can be done in FPT time parameterized by k in total,since each component has at most k vertices and the number of components in G − W is at most | V | . We now check whether by picking one pair ( p, q ) for eachcomponent, it is possible to make the total number of components r − k (cid:48) . Thiscan be done in polynomial time by a standard dynamic programming algorithmsince the number of components and r − k (cid:48) are at most | V | . (cid:117)(cid:116) F Hard problems parameterized by vi F.1 Graph Motif
Given a graph G = ( V, E ), a vertex coloring c : V → C , and a multiset M ofcolors in C , the problem Graph Motif is to decide if there is a set S ⊆ V such that G [ S ] is connected and c ( S ) = M , where c ( S ) is the multiset of colorsappearing in S . If the motif M is a set (i.e., no element in M has multiplicity morethan 1), then the restricted problem is called Colorful Graph Motif . It isknown that
Graph Motif is fixed-parameter tractable parameterized by vc [14](actually by more general parameters neighborhood diversity [32] and twin-covernumber [31]). The proof of Theorem 20 in [14] implies that Colorful GraphMotif is NP-complete for graphs of vi = 6. By a similar proof, we will showthat Colorful Graph Motif is NP-complete for graphs of vi = 4. We thencomplement this by showing that Graph Motif is polynomial-time solvable forgraphs of vi ≤ Theorem F.1.
Colorful Graph Motif is NP-complete on trees of vertexintegrity .Proof. The problem is clearly in NP. We present a reduction from an NP-complete problem [34]. The input of consists of three disjoint sets X = { x , . . . , x n } , Y = { y , . . . , y n } , Z = { z , . . . , z n } , and a set of triples T ⊆ X × Y × Z . The task is to decidewhether there is a subset S of T such that | S | = n and each element of X ∪ Y ∪ Z appears in a triple included in S .We construct a graph G with a coloring c as follows. The graph G contains aspecial root vertex r with unique color c ( r ) = r . For each triple t = ( x i , y j , z k ) ∈ T , take three new vertices t i , t j , t k with c ( t i ) = x i , c ( t j ) = y j , and c ( t k ) = z k , andadd three new edges { r, t i } , { t i , t j } , and { t j , t k } . We set M = { r } ∪ X ∪ Y ∪ Z .This completes the construction. Note that G is a tree and { r } is a vi (4)-set of G as each component of G − { r } is a path of order 3.Assume that ( X, Y, Z, T ) is a yes instance of with S ⊆ T as a certificate. We set L = { r } ∪ { t i , t j , t k | t = ( x i , y j , z k ) ∈ S } . Clearly, c ( L ) = M . Since G [ L ] is connected, ( G, M ) is a yes instance of
Colorful Graph Motif . reedepth, Vertex Cover, and Vertex Integrity 27 To show the other direction, assume that a vertex subset L of G induces aconnected graph and c ( L ) = M . Observe that L has to include r . Since X ∪ Y ∪ Z = c ( L ) \ { r } , L includes exactly n vertices of distance i from r for each i ∈ { , , } . This fact and the connectivity of G [ L ] imply that for each t =( x i , y j , z k ) ∈ T , L contains either all vertices t i , t j , t k or none of them. Let S ⊆ T be the set of triples such that L contains all three vertices correspondingto each t ∈ S . By the discussion above, | S | = n and each element of X ∪ Y ∪ Z appears in a triple included in S . (cid:117)(cid:116) Theorem F.2.
Graph Motif can be solved in polynomial time on graphs ofvertex integrity at most .Proof. Let G = ( V, E ) be the input graph with a coloring c : V → C and M be the input multiset of colors. Let R be a vi (3)-set of G . If | R | ≥
2, then R is a vertex cover of G with | R | ≤
3, and thus we can apply an FPT algorithmparameterized by the vertex cover number [31,14]. If R = ∅ , then each connectedcomponent of G is of order at most 3, and thus the problem is trivial. In thefollowing we assume that R = { r } for some r ∈ V . Furthermore, we assume that r is included in the solution S as otherwise | S | ≤
2. Let D i be the vertices ofdistance i from r . Note that V = { r } ∪ D ∪ D .We construct an auxiliary bipartite multi-graph H as follows. For each color x ∈ C , take new vertices x and x . For each component C of G − r , if C has twovertices u of color x and v of color y , where only u is adjacent to r , then addone edge between x and y . For each color x ∈ C , we define degree constraintsof x and x in H as follows: the degree constraint of x is “at most M ( x )”,and the degree constraint of x is “exactly max { M ( x ) − q ( x ) , } ”, where M ( x )is the multiplicity of x in M \ { c ( r ) } and q ( x ) is the number of color- x verticesin D . We will show that H has a subgraph F satisfying the degree constraintsif and only if there is a set S ⊆ V such that G [ S ] is connected and c ( S ) = M .Since finding a subgraph of such degree constraints can be done in polynomialtime [30], this equivalence implies the theorem.First assume that there is a set S ⊆ V such that G [ S ] is connected and c ( S ) = M . We choose S among such sets so that | S ∩ D | is maximized. Thisimplies in particular that if there is a vertex v ∈ D \ S of color x , then novertex of color x in D belongs to S . For each edge { u, v } in G [ S − r ], if u ∈ D , v ∈ D , c ( u ) = x , and c ( v ) = y , then add one edge between x and y into F . Now for each color x , the degree of x in F is at most M ( x ). Since S takescolor- x vertices in D only when it is necessary after including all color- x verticesin D , the degree of x in F is exactly max { M ( x ) − q ( x ) , } .Next assume that H has a subgraph F that satisfies the degree constraints.For each edge between x and y in F , we add into S the endpoints of anarbitrary edge { u, v } in G − r such that c ( u ) = x , u ∈ D , c ( v ) = y , and v ∈ D . Let S = c ( S ), the multiset of colors appear in S . From the constructionof S , it holds for each color x that S ( x ) = deg F ( x ) + deg F ( x ) = deg F ( x ) +max { M ( x ) − q ( x ) , } . If M ( x ) ≤ q ( x ), then S ( x ) = deg F ( x ) ≤ M ( x ). We add,into S , arbitrary M ( x ) − S ( x ) of color- x vertices in D \ S . This is possible since the number of color- x vertices in D \ S is q ( x ) − S ( x ) ≥ M ( x ) − S ( x ).If M ( x ) > q ( x ), then S ( x ) = deg F ( x ) + M ( x ) − q ( x ). In this case, we add allcolor- x vertices in D into S , and then the multiplicity of x in the resultant setbecomes q ( x ) + M ( x ) − q ( x ) = M ( x ). (cid:117)(cid:116) F.2 Steiner Forest
Steiner Forest is a generalization of
Steiner Tree and defined as follows:Given a graph G = ( V, E ) with edge weighting w : E → Z + , a positive integer k ,and disjoint terminal sets T , . . . , T t ⊆ V with | T i | ≥ i , decide whetherthere is a subgraph F of G with (cid:80) e ∈ F w ( e ) ≤ k such that each T i is containedin some connected component of F . Note that we can assume that F is a forest.It is known that Steiner Forest is strongly NP-complete (that is, NP-complete even if the weights are given in unary) on graphs of vertex integrity5 [35]. We show that for graphs of small vertex cover number, the problembecomes easier.Let G = ( V, E ) be a graph, F a subgraph of G , and S a vertex cover of G . Weassume without loss of generality that F is a forest. The following observationsfollow from this assumption and the fact that V − S is an independent set. Observation F.3
At most | S | − vertices in V − S have degree 2 or more in F . Observation F.4 F has at most | S | connected components. Theorem F.5.
Steiner Forest can be solved in time n O ( vc ) , on n -vertexgraphs of vertex cover number at most vc .Proof. Let G = ( V, E ) be a graph and S ⊆ V be a vertex cover of G . Wefirst guess the set D ⊆ V − S of vertices that have degree at least 2 in F . ByObservation F.3, we know that | D | ≤ | S | −
1. The vertices in V − ( S ∪ D ) havedegree at most 1 in F : if such a vertex appears in some T i , then it has degree 1in F ; otherwise, it has degree 0 in F , and thus can be safely removed from thegraph. We then guess the edges in F [ S ∪ D ]. The number of candidates for D isat most n | S | , and the number of candidates for the edge set is at most | S | | S | .We reject the guess F [ S ∪ D ] if there are two components of F [ S ∪ D ] thatcontain elements of T i for some i . Now for each i , there is at most one component C i of F [ S ∪ D ] such that C ∩ T i (cid:54) = ∅ . If there is no such component, we guess onecomponent of F [ S ∪ D ] from O ( | S | ) candidates and call it C i . Note that C i and C j may be the same for i (cid:54) = j . Now for each i and for each vertex u ∈ T i \ ( S ∪ D ),we find an edge { u, v } of the minimum weight such that v ∈ V ( C i ) and add { u, v } into F . We output a minimum weight forest obtained in this way. (cid:117)(cid:116) We denote by
Unweighted Steiner Forest the special case of
SteinerForest such that each edge has weight 1. By subdividing the edges in the proofin [35], we can show that
Unweighted Steiner Forest is NP-complete forgraphs of tw = 3. reedepth, Vertex Cover, and Vertex Integrity 29 Theorem F.6.
Unweighted Steiner Forest is fixed-parameter tractableparameterized by vc .Proof. Let G = ( V, E ) be the input graph and S be a vertex cover of G . Let s = | S | . We reduce the instance by applying the following reduction rules ex-haustively.1. If T i contains s or more vertices v in V − S that have the same neighborhood N ( v ), then remove one of them from T i and decrease k by 1.2. Let T i (1) , . . . T i ( s +1) be s + 1 distinct terminal sets such that T i ( j ) ∩ S = ∅ for all j and for each X ⊆ S and j (cid:54) = j (cid:48) , it holds that |{ v ∈ T i ( j ) | N ( v ) = X }| = |{ v ∈ T i ( j (cid:48) ) | N ( v ) = X }| . Then replace T i (1) and T i (2) with theirunion T i (1) ∪ T i (2) .3. If there are two non-terminal vertices of the same neighborhood, then removeone of them from the graph.The first rule is safe by Observation F.3. At least one of the s vertices of thesame neighborhood is a leaf. Since there are other vertices of the same neighbor-hood in T i , this leaf can be joined to the component containing the other verticesof T i with an edge. The safeness of the second rule follows by Observation F.4,as at least two of them belong to the same connected component. The third ruleis safe because at most one of them is used in an optimal solution.We can see that if no reduction rule above applies, then the numbers ofvertices and of terminal sets depend only on s . By the first rule, each terminalset T i contains at most s · s vertices. By the first and second rules, there areat most s · s s + s terminal sets. By Reduction rule 3, there are at most 2 s + s non-terminal vertices. (cid:117)(cid:116) G Easy problems parameterized by td Here we show the following.
Observation G.1
List Hamiltonian Path , Directed ( p, q ) - Edge Domi-nating Set , and
Metric Dimension are fixed-parameter tractable parameter-ized by td . List Hamiltonian Path is a generalization of
Hamiltonian Path suchthat each vertex has a set of permitted positions where it can be put in a Hamil-tonian path. This problem is W[1]-hard parameterized by pw [43]. By Proposi-tion A.1, this problem parameterized by td admits a trivial FPT algorithm: if agraph G has at least 2 td ( G ) vertices, then G does not have a Hamiltonian path;otherwise, we can try all n ! ≤ (2 td ( G ) )! permutations of vertices.For integers p, q ≥
0, the edges ( p, q ) -dominated by an arc e = ( u, v ) are e itself and all arcs that are on a directed path of length at most p to u or on adirected path of length at most q from v . Then, Directed ( p, q )- Edge Domi-nating Set asks whether there exists a set K of arcs with | K | ≤ k such thatevery arc is ( p, q )-dominated by K . This problem is W[1]-hard parameterized by pw but fixed-parameter tractable parameterized by tw + p + q [7]. Since we canassume that p and q are smaller than the longest path length, we can also assumethat they are bounded by a function of td . Thus the fixed-parameter tractabilitywith tw + p + q implies the fixed-parameter tractability solely with td . (Herethe parameters are defined on the undirected graph obtained by ignoring thedirections of arcs.)The argument for Metric Dimension is slightly more involved. In
MetricDimension , we are given a graph G = ( V, E ) and an integer k and asked whetherthere exists S ⊆ V such that | S | ≤ k and for each pair u, v ∈ V there exists w ∈ S with dist( u, w ) (cid:54) = dist( v, w ), where dist( · , · ) is the distance between two verticesin G . We call such a set S a resolving set . Recently, this problem is shown tobe W[1]-hard parameterized by pw [13]. Observe that there is an MSO formula ϕ ( S ) such that its length depends only on the diameter of the underlying graphand it is evaluated to be true if and only if the vertex set S is a resolving set of theunderlying graph. It is known that finding a minimum vertex set S satisfying ϕ ( S ) is fixed-parameter tractable parameterized by clique-width + | ϕ | [18,48].Therefore, Metric Dimension is fixed-parameter tractable parameterized byclique-width+the diameter. Since the clique-width and the diameter are boundedfrom above by functions of its treedepth,
Metric Dimension is fixed-parametertractable parameterized by tdtd