Solving Problems on Generalized Convex Graphs via Mim-Width
Flavia Bonomo-Braberman, Nick Brettell, Andrea Munaro, Daniël Paulusma
SSolving problems on generalized convex graphs viamim-width
Nick Brettell
School of Mathematics and Statistics, Victoria University of Wellington, New [email protected]
Andrea Munaro
School of Mathematics and Physics, Queen’s University Belfast, [email protected]
Daniël Paulusma
Department of Computer Science, Durham University, [email protected]
Abstract
A bipartite graph G = ( A, B, E ) is H -convex, for some family of graphs H , if there exists a graph H ∈ H with V ( H ) = A such that the set of neighbours in A of each b ∈ B induces a connected subgraph of H . A variety ofwell-known NP -complete problems, including Dominating Set , Feedback Vertex Set , Induced Matching and
List k -Colouring , become polynomial-time solvable for H -convex graphs when H is the set of paths. Inthis case, the class of H -convex graphs is known as the class of convex graphs. The underlying reason is that theclass of convex graphs has bounded mim-width. We extend the latter result to families of H -convex graphs where(i) H is the set of cycles, or(ii) H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3.As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs knownin the literature. To complement result (ii), we show that the mim-width of H -convex graphs is unbounded if H isthe set of trees with arbitrarily large maximum degree or arbitrarily large number of vertices of degree at least 3.In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Theory of computation → Graph algorithms analysis
Keywords and phrases convex, circular convex, tree convex, width parameter, mim-width, complexity dichotomy
Funding
The research in this paper received support from the Leverhulme Trust (RPG-2016-258).
Many computationally hard graph problems can be solved efficiently if we place constraints on the input.Instead of trying to apply this approach to individual problems in an ad hoc way, it is more insightful toseek the underlying reason why some graph problems are easier on certain classes of input than others. Ageneral method towards this goal is to try to decompose the vertex set of the input graph into large sets of“similarly behaving” vertices and to exploit this decomposition for an algorithmic speed up that works formany problems simultaneously. This requires some notion of an “optimal” vertex decomposition, whichdepends on the type of vertex decomposition used and which may relate to the minimum number of setsor the maximum size of a set in a vertex decomposition. An optimal vertex decomposition gives us the“width” of the graph. A graph class has bounded width if every graph in the class has width at most someconstant c . Boundedness of width is often the underlying reason why a graph-class-specific algorithm runsefficiently: in such a case, the proof that the algorithm is efficient for some special graph class reduces toa proof showing that the width of the class is bounded by some constant. We will see illustratations ofthis here , but also refer to the surveys [23, 29, 30, 41, 60] for further details and examples.Various width parameters differ in strength. To explain this, a width parameter p dominates a widthparameter q if there is a function f such that p ( G ) is at most f ( q ( G )) for every graph G . If p dominates q but q does not dominate p , then we say that p is more powerful than q . If both p and q dominate each a r X i v : . [ c s . D S ] A ug Solving problems on generalized convex graphs via mim-width other, then p and q are equivalent . If p is more powerful than q , then the class of graphs for which p isbounded is larger than the class of graphs for which q is bounded and so efficient algorithms for bounded p have greater applicability with respect to the graphs under consideration. The trade-off is that fewerproblems exhibit an efficient algorithm for the parameter p , compared to the parameter q .The notion of powerfulness leads to a large hierarchy of width parameters, and new width parameterscontinue to be defined, such as graph functionality [1] in 2019 and twin-width [5] in 2020. The well-knownparameters boolean-width, clique-width, module-width, NLC-width and rank-width are all equivalent toeach other [14, 39, 49, 54]. They are more powerful than the equivalent parameters branch-width andtreewidth [21, 55, 60] but less powerful than mim-width [60], which is less powerful than sim-width [42].For each group of equivalent width parameters, a growing set of NP -complete problems is known to betractable on graph classes of bounded width. Proving boundedness of width of some graph class oftenimmediately tells us that many problems are tractable for that class without the need for constructingalgorithms for each problem. Hence, it is natural to ask which structural properties of a graph class ensureconstant-bounded width. Our Focus.
There are large families of natural graph classes for which the (un)boundedness of widthis not known for many width parameters. We will focus on a relatively new width parameter, namelythe aforementioned parameter mim-width , which we define below. Recently, we showed in [11, 12]that boundedness of mim-width is the underlying reason why some specific hereditary graph classes,characterized by two forbidden induced subgraphs, admit polynomial-time algorithms for a range ofproblems including k -Colouring and its generalization List k -Colouring (the algorithms are givenin [18, 22, 31]). In this paper we prove that the same holds for certain superclasses of convex graphs that have been studied in the literature. Essentially all the known polynomial-time algorithms for suchclasses are obtained by reducing to the class of convex graphs. We show that our approach via mim-widthsimplifies the analysis, unifies the sporadic approaches and explains the reductions to convex graphs. A set of edges M in a graph G is a matching if no two edges of M share an endpoint. A matching M is induced if there is no edge in G between vertices of different edges of M . Let ( A, A ) be a partition of thevertex set of a graph G . Then G [ A, A ] denotes the bipartite subgraph of G induced by the edges with oneendpoint in A and the other in A . Motivated by algorithmic applications, Vatshelle [60] introduced thenotion of maximum induced matching width , also called mim-width. Mim-width measures the extent towhich it is possible to decompose a graph G along certain vertex partitions ( A, A ) such that the size ofa maximum induced matching in G [ A, A ] is small. The kind of vertex partitions permitted stem fromclassical branch decompositions. A branch decomposition for a graph G is a pair ( T, δ ), where T is asubcubic tree and δ is a bijection from V ( G ) to the leaves of T . Every edge e ∈ E ( T ) partitions the leavesof T into two classes, L e and L e , depending on which component of T − e they belong to. Hence, e inducesa partition ( A e , A e ) of V ( G ), where δ ( A e ) = L e and δ ( A e ) = L e . Let cutmim G ( A e , A e ) be the size of amaximum induced matching in G [ A e , A e ]. Then the mim-width mimw G ( T, δ ) of (
T, δ ) is the maximumvalue of cutmim G ( A e , A e ) over all edges e ∈ E ( T ). The mim-width mimw( G ) of G is the minimum valueof mimw G ( T, δ ) over all branch decompositions (
T, δ ) for G . We refer to Figure 1 for an example.Computing the mim-width is an NP -hard problem [56]. Moreover, approximating the mim-width inpolynomial time within a constant factor of the optimal is not possible unless NP = ZPP [56]. Currently,it is not known, in general, how to compute in polynomial time a branch decomposition for a graph G whose mim-width is bounded by some function in the mim-width of G . Hence, for a class of graphs G of bounded mim-width, we need a polynomial-time algorithm for computing a branch decompositionwhose mim-width is bounded by a constant. If this is possible, the mim-width of G is said to be quicklycomputable . One can then try to develop a polynomial-time algorithm for the graph problem under . Brettell, A. Munaro, and D. Paulusma 3 consideration via dynamic programming over the computed branch decomposition. We briefly discuss anumber of graph problems for which this turned out to be possible.First of all, Belmonte and Vatshelle [2] and Bui-Xuan et al. [15] proved that the so-called LocallyCheckable Vertex Subset and Vertex Partitioning (LC-VSVP) problems, first defined in [59], are polynomial-time solvable on graph classes whose mim-width is bounded and quickly computable. Examples ofsuch problems are (Dominating) Induced Matching , (Total) Dominating Set , IndependentDominating Set , Independent Set and k - Colouring for every fixed positive integer k . Kwon [43]observed that the same holds for List k -Colouring for every fixed k (see [11] for details). Jaffke et al.[34] showed that the distance versions of LC-VSVP problems can also be solved in polynomial time forgraph classes whose mim-width is bounded and quickly computable. Jaffke et al. [35, 36] proved thesame for Longest Induced Path , Induced Disjoint Paths , H - Induced Topological Minor and
Feedback Vertex Set . Bergougnoux et al. [4] generalized the result for
Feedback Vertex Set to Weighted Subset Feedback Vertex Set and
Weighted Node Multiway Cut . Bergougnoux andKanté [3] proved that
Connected Dominating Set , Node Weighted Steiner Tree and
MaximumInduced Tree are all polynomial-time solvable on graph classes whose mim-width is bounded and quicklycomputable. Galby et al. [26] proved the same for
Semitotal Dominating Set , whereas Chudnovskyet al. [19] observed this for
Max Partial H -Colouring , a problem that generalizes, amongst others, Odd Cycle Transversal . A bipartite graph G = ( A, B, E ) is convex if there exists a path P with V ( P ) = A such that the neighboursin A of each b ∈ B induce a connected subpath of P . Convex graphs generalize bipartite permutationgraphs (see, e.g., [8]) and form a well-studied graph class. They were introduced in the sixties, by Glover[27], to solve a special type of matching problem arising in some industrial application. Another earlypaper solving matching problems on convex graphs is by Lipski Jr. and Preparata [40].The clique-width of bipartite permutation graphs, and hence convex graphs, is unbounded [7]. However, a a a a a a a a b b b b (a) b b a a b a b a a a a a a a a a a b b a a b a b e (b)Figure 1 (a) A circular convex graph G = ( A, B, E ) with a circular ordering on A . (b) A branch decomposition( T, δ ) for G , where T is a caterpillar with a specified edge e , together with the graph G [ A e , A e ]. The bold edges in G [ A e , A e ] form an induced matching and it is easy to see that cutmim G ( A e , A e ) = 2. Solving problems on generalized convex graphs via mim-width
Belmonte and Vatshelle [2] proved that the mim-width of convex graphs is bounded and quickly computable.This implies that all the aforementioned graph problems are polynomial-time solvable on convex graphs,providing alternative proofs for the already known polynomial-time results (see, e.g., [24]). To giveanother very recent example, earlier in 2020 Díaz et al. [25] provided a polynomial-time algorithm for
List k -Colouring on convex graphs. Now that we know that List k -Colouring is polynomial-timesolvable for any class of graphs whose mim-width is bounded and quickly computable [43] (see [11]), apolynomial-time algorithm also directly follows from the result of [2]. In the remainder of our paper, weconsider superclasses of convex graphs and research to what extent mim-width can play a role in obtainingpolynomial-time algorithms for problems on these classes.Let H be a family of graphs. A bipartite graph G = ( A, B, E ) is H -convex if there exists a graph H ∈ H with V ( H ) = A such that the set of neighbours in A of each b ∈ B induces a connected subgraph of H . If H consists of all paths, we obtain the class of convex graphs. A caterpillar is a tree T that contains a path P , called the backbone of T , such that every vertex not on P has a neighbour on P . A caterpillar with abackbone consisting of one vertex is a star . A comb is a caterpillar such that every backbone vertex hasexactly one neighbour outside the backbone. The subdivision of an edge uv replaces uv by a new vertex w and edges uw and wu . A triad is a tree that can be obtained from a 4-vertex star after a sequence ofsubdivisions. For non-negative integers t and ∆, a ( t, ∆) -tree is a tree with maximum degree at most ∆and containing at most t vertices of degree at least 3; note that, for example, a triad is a (1 , H consists of all cycles, all trees, all stars, all triads, all combs or all ( t, ∆)-trees, then we obtain theclass of circular convex graphs , tree convex graphs , star convex graphs , triad convex graphs , comb convexgraphs or ( t, ∆) -tree convex graphs , respectively. See Figure 1 for an example of a circular convex graph(this class was introduced by Liang and Blum [44] to model certain scheduling problems).To show the relationships between the above graph classes we need some extra terminology. Let C t, ∆ bethe class of ( t, ∆)-tree convex graphs. For fixed t or ∆, we have increasing sequences C t, ⊆ C t, ⊆ · · · and C , ∆ ⊆ C , ∆ ⊆ · · · . For t ∈ N , the class of ( t, ∞ ) -tree convex graphs is S ∆ ∈ N C t, ∆ , denoted by C t, ∞ .Similarly, for ∆ ∈ N , the class of ( ∞ , ∆) -tree convex graphs is S t ∈ N C t, ∆ , denoted by C ∞ , ∆ . Hence, C t, ∞ and C ∞ , ∆ are the set-theoretic limits of the increasing sequences {C t, ∆ } ∆ ∈ N and {C t, ∆ } t ∈ N , respectively.The class of ( ∞ , ∞ ) -tree convex graphs is S t, ∆ ∈ N C t, ∆ , which coincides with the class of tree convex graphs.Notice that the class of convex graphs coincides with C t, , for any t ∈ N ∪ {∞} , and with C , ∆ , for any∆ ∈ N ∪ {∞} . The class of star convex graphs coincides with C , ∞ . Moreover, each triad convex graphbelongs to C , and each comb convex graph belongs to C ∞ , . A bipartite graph is chordal bipartite ifevery induced cycle in it has exactly four vertices. Every convex graph is chordal bipartite (see, e.g., [8])and every chordal bipartite graph is tree convex (see [38, 45]). In Figure 2 we display these and otherrelationships, which directly follow from the definitions.Brault-Baron et al. [9] proved that chordal bipartite graphs have unbounded mim-width and so theresult of Belmonte and Vatshelle [2] for convex graphs cannot be generalized to chordal bipartite graphs.In our paper we determine the mim-width of the other classes in Figure 2, but first we discuss knownalgorithmic results for these classes. Each of the problems mentioned below is a special case of a Locally Checkable Vertex Subset (LCVS)problem. We refer to the listed papers for their definitions, as we do not need these definitions here.Panda et al. [51] proved that
Induced Matching is polynomial-time solvable for circular convexgraphs and triad convex graphs, but NP -complete for star convex and comb convex graphs. Pandey andPanda [52] proved that Dominating Set is polynomial-time solvable for circular convex, triad convexand (1 , ∆)-tree convex graphs for every ∆ ≥
1. Liu et al. [47] proved that
Connected Dominating Set is polynomial-time solvable for circular convex graphs and triad convex graphs. Chen et al. [16] showedthat
Dominating Set , Connected Dominating Set and
Total Dominating Set are NP -complete . Brettell, A. Munaro, and D. Paulusma 5 bipartitetree convex ≡ ( ∞ , ∞ )-tree convex circular convex( t, ∆)-tree convex, t ≥ , ∆ ≥ ∞ , ≡ (1 , ∞ )-tree convex chordal bipartitetriad convexconvex Figure 2
The inclusion relations between the classes mentioned in our paper. A line from a lower-level classto a higher one indicates that the first class is contained in the second. The dotted line separates the classes ofbounded mim-width and unbounded mim-width (see also Theorems 1–3). for star convex and comb convex graphs.Lu et al. [48] proved that
Independent Dominating Set is polynomial-time solvable for circularconvex and triad convex graphs. The latter result was shown already by Song et al. [58] who used adynamic programming approach instead of a reduction to convex graphs, as done in [48]. Song et al. [58]showed in fact a stronger result, namely that
Independent Dominating Set is polynomial-time solvablefor ( t, ∆)-tree convex graphs for every t ≥ ≥
3. They also showed in [58] that
IndependentDominating Set is NP -complete for star convex graphs and for comb convex graphs. Hence, theyobtained a dichotomy: Independent Dominating Set is polynomial-time solvable for ( t, ∆)-tree convexgraphs for every t ≥ ≥ NP -complete for ( ∞ , , ∞ )-treeconvex graphs.The same dichotomy (explicitly formulated in [61]) holds for Feedback Vertex Set and is obtainedsimilarly. Namely, Jiang et al. [37] proved that this problem is polynomial-time solvable for triad convexgraphs and mentioned that their algorithm can be generalized to ( t, ∆)-tree convex graphs for every t ≥ ≥
3. Jiang et al. [38] proved that
Feedback Vertex Set is NP -complete for star convex andcomb convex graphs. In addition, Liu et al. [46] proved that Feedback Vertex Set is polynomial-timesolvable for circular convex graphs, whereas Jiang et al. [38] proved that the
Weighted FeedbackVertex Set problem is polynomial-time solvable for triad convex graphs.It turns out that the above problems are polynomial-time solvable on circular convex graphs andsubclasses of ( t, ∆)-tree convex graphs, but NP -complete for star convex graphs and comb convex graphs(and for two problems this led to a dichotomy result). In contrast, Panda and Chaudhary [50] provedthat Dominating Induced Matching is not only polynomial-time solvable on circular convex andtriad convex graphs, but also on star convex graphs. Nevertheless, we notice a common pattern: manydominating set, induced matching and graph transversal type of problems are polynomial-time solvablefor ( t, ∆)-tree convex graphs, for every t ≥ ≥
3, and NP -complete for comb convex graphs, and Solving problems on generalized convex graphs via mim-width thus for ( ∞ , , ∞ )-tree convex graphs.Moreover, essentially all the polynomial-time algorithms reduce the input to a convex graph. As mentioned, our goal is to simplify the analysis, unify the approaches in Section 1.3 and explain thereductions to convex graphs, using mim-width.
Structural Results.
We prove the following results explaining the dotted line in Figure 2. The first tworesults generalize that of Belmonte and Vatshelle [2], as convex graphs form a common subclass of circularconvex graphs and (1 , ∞ , ∞ )-tree convex graphs) have unbounded mim-width. (cid:73) Theorem 1.
Let G be a circular convex graph. Then mimw( G ) ≤ . Moreover, we can construct inpolynomial time a branch decomposition ( T, δ ) for G with mimw G ( T, δ ) ≤ . (cid:73) Theorem 2.
Let G be a ( t, ∆) -tree convex graph with t, ∆ ∈ N and t ≥ and ∆ ≥ . Let f ( t, ∆) = max ( $(cid:18) ∆2 (cid:19) % , − ) + 2 t − ∆ . Then mimw( G ) ≤ f ( t, ∆) . Moreover, we can construct in polynomial time a branch decomposition ( T, δ ) for G with mimw G ( T, δ ) ≤ f ( t, ∆) . (cid:73) Theorem 3.
The classes of star convex graphs and comb convex graphs have unbounded mim-width.
As a consequence, we obtain the following structural dichotomy (recall that star convex graphs are the(1 , ∞ )-tree convex graphs and that comb convex graphs form a subclass of ( ∞ , (cid:73) Corollary 4.
Let t, ∆ ∈ N ∪ {∞} with t ≥ and ∆ ≥ . The class of ( t, ∆) -tree convex graphs hasbounded mim-width if and only if { t, ∆ } ∩ {∞} = ∅ . Algorithmic Consequences.
As explained in Section 1.3, the following six problems were shown tobe NP -complete for both star convex graphs and comb convex graphs, and thus for (1 , ∞ )-tree convexgraphs and ( ∞ , Feedback Vertex Set [38];
Dominating Set , ConnectedDominating Set , Total Dominating Set [16];
Independent Dominating Set [58];
InducedMatching [51]. All these problems are examples of Locally Checkable Vertex Subset (LCVS) problems.Hence, they are polynomial-time solvable for every graph class whose mim-width is bounded and quicklycomputable [15]. We also recall that the same result holds for
Weighted Feedback Vertex Set [36]and (Weighted) Subset Feedback Vertex Set [4]; each of these problems generalizes
FeedbackVertex Set and is thus NP -complete for star convex graphs and comb convex graphs. Combining theseresults with Corollary 4 yields the following complexity dichotomy. (cid:73) Corollary 5.
Let t, ∆ ∈ N ∪ {∞} with t ≥ , ∆ ≥ and let Π be one of the nine problems mentionedabove. If { t, ∆ } ∩ {∞} = ∅ , then Π is polynomial-time solvable when restricted to ( t, ∆) -tree convexgraphs. Otherwise, Π is NP -complete. It is worth noting that this complexity dichotomy does not hold for all LCVS problems; recall that
Dominating Induced Matching is polynomial-time solvable on star convex graphs [50]. Theorems 1and 2, combined with the result of [15], imply that this problem is also polynomial-time solvable on circularconvex graphs and ( t, ∆)-tree convex graphs for every t ≥ ≥
3. On another note, Theorems 1and 2, combined with the result of [43], also generalize the aforementioned result of Díaz et al. [25] for
List k -Colouring on convex graphs to circular convex graphs and ( t, ∆)-tree convex graphs ( t ≥ ≥ . Brettell, A. Munaro, and D. Paulusma 7 We consider only finite graphs G = ( V, E ) with no loops and no multiple edges. For a vertex v ∈ V , the neighbourhood N ( v ) is the set of vertices adjacent to v in G . The degree d ( v ) of a vertex v ∈ V is the size | N ( v ) | of its neighbourhood. A vertex of degree k is a k -vertex . A graph is subcubic if every vertex hasdegree at most 3. We let ∆( G ) = max { d ( v ) : v ∈ V } . The distance d G ( u, v ) from a vertex u to a vertex v is the length of a shortest path between u and v in G . For disjoint S, T ⊆ V , we say that S is complete to T if every vertex of S is adjacent to every vertex of T . For S ⊆ V , G [ S ] = ( S, { uv : u, v ∈ S, uv ∈ E } ) isthe subgraph of G induced by S . The disjoint union G + H of graphs G and H has vertex set V ( G ) ∪ V ( H )and edge set E ( G ) ∪ E ( H ). For a graph H , a graph G is H -free if G has no induced subgraph isomorphicto H . The path on n vertices is denoted by P n . A graph is r -partite , for r ≥
2, if its vertex set admits apartition into r classes such that every edge has its endpoints in different classes. A 2-partite graph isalso called bipartite . A graph G is a support for a hypergraph H = ( V, S ) if the vertices of G correspondto the vertices of H and, for each hyperedge S i ∈ S , the subgraph of G induced by S i is connected. So,support graphs are witnesses for proving that a certain graph is H -convex for some family of graphs H . In this section we prove Theorem 1. We need the following known lemma on recognizing circular convexgraphs. (cid:73)
Lemma 6 (see, e.g., Buchin et al. [13]) . Circular convex graphs can be recognized and a cycle supportcomputed, if it exists, in polynomial time.
For an integer ‘ ≥
1, an ‘ -caterpillar is a subcubic tree T on 2 ‘ vertices with V ( T ) = { s , . . . , s ‘ , t , . . . , t ‘ } ,such that E ( T ) = { s i t i : 1 ≤ i ≤ ‘ } ∪ { s i s i +1 : 1 ≤ i ≤ ‘ − } . Note that we label the leaves of an ‘ -caterpillar t , t , . . . , t ‘ , in this order. Given a total ordering ≺ of length ‘ , we say that ( T, δ ) is obtainedfrom ≺ if T is an ‘ -caterpillar and δ is the natural bijection from the ‘ ordered elements to the leaves of T .We are now ready to prove Theorem 1. (cid:73) Theorem 1 (restated).
Let G be a circular convex graph. Then mimw( G ) ≤ . Moreover, we canconstruct in polynomial time a branch decomposition ( T, δ ) for G with mimw G ( T, δ ) ≤ . Proof.
Let G = ( A, B, E ) be a circular convex graph with a circular ordering on A . By Lemma 6, weconstruct in polynomial time such an ordering a , . . . , a n , where n = | A | (see Figure 1). Let B = { b ∈ B : a n ∈ N ( b ) } and B = B \ B . We obtain a total ordering ≺ on V ( G ) by extending the ordering a , . . . , a n as follows. Each b ∈ B is inserted after a n , breaking ties arbitrarily. Each b ∈ B is inserted immediatelyafter the largest element of A it is adjacent to (hence immediately after some a i with 1 ≤ i < n ), breakingties arbitrarily.Let T be the | V ( G ) | -caterpillar obtained from ≺ . We show that mimw G ( T, δ ) ≤
2. Let e be an edgeof T and let M be a maximum induced matching of G [ A e , A e ], where each vertex in A e is larger than anyvertex in A e . Observe first that at most one edge of M has one endpoint in B . Indeed, suppose thereexist two edges xy, x y ∈ M , each with one endpoint in B , say without loss of generality { y, y } ⊆ B .Since each vertex in B is adjacent only to smaller vertices, { y, y } ⊆ A e and { x, x } ⊆ A e . Withoutloss of generality, y ≺ y . But N ( y ) and N ( y ) are intervals of the ordering and so either x ∈ N ( y )or x ∈ N ( y ), contradicting the fact that M is induced. We finally show that at most two edges in M have an endpoint in B and, if exactly two such edges are in M , then no edge with an endpoint in B is. Suppose, to the contrary, that three edges of M have one endpoint in B and let u , u , u be theseendpoints. Since N ( u ), N ( u ) and N ( u ) are intervals of the circular ordering on A all containing a n ,one of these neighbourhoods is contained in the union of the other two, contradicting the fact that M Solving problems on generalized convex graphs via mim-width is induced. Suppose finally that exactly two edges u v , u v ∈ M have one endpoint in B . We mayassume { u , u } ⊆ A e , { v , v } ⊆ A e and u ∈ B . Then u ∈ B and so { v , v } ⊆ A . Now if there issome edge u v ∈ M such that u ∈ B , then u ∈ A e . Recall that N ( u ) and N ( u ) are intervals ofthe circular ordering on A both containing a n . Since M is induced, for each i, j ∈ { , } , we have that v i ∈ N ( u j ), if i = j , and v i / ∈ N ( u j ), if i = j . This implies that one of v and v is larger than v in ≺ and so it is contained in N ( u ), contradicting the fact that M is induced. This concludes the proof. (cid:74) In this section we prove Theorem 2. We need the following lemma on recognizing ( t, ∆)-tree convexgraphs . (cid:73) Lemma 7.
For t, ∆ ∈ N , ( t, ∆) -tree convex graphs can be recognized and a ( t, ∆) -tree support computed,if it exists, in polynomial time. Proof.
Given a hypergraph H = ( V, S ) together with degrees d i for each i ∈ V , Buchin et al. [13] providedan O ( | V | + |S|| V | ) time algorithm that solves the following decision problem: Is there a tree supportfor H such that each vertex i of the tree has degree at most d i ? If it exists, the algorithm computes atree support satisfying this property. Given as input a bipartite graph G = ( A, B, E ), we consider thehypergraph H = ( A, S ), where S = { N ( b ) : b ∈ B } . For each of the (cid:0) | A | t (cid:1) = O ( | A | t ) subsets A ⊆ A ofsize t we proceed as follows: we assign a degree ∆ to each of its elements and a degree 2 to each element in A \ A . We then apply the algorithm in [13] to the O ( | A | t ) instances thus constructed. If G is a ( t, ∆)-treeconvex graph, the algorithm returns a ( t, ∆)-tree support for H . (cid:74) The proof of Theorem 2 heavily relies on the following general result for mim-width. (cid:73)
Lemma 8 (Brettell et al. [11]) . Let G be a graph and ( X , . . . , X p ) be a partition of V ( G ) such that cutmim G ( X i , X j ) ≤ c for all distinct i, j ∈ { , . . . , p } , and p ≥ . Let h = max (cid:26) c (cid:22)(cid:16) p (cid:17) (cid:23) , max i ∈{ ,...,p } { mimw( G [ X i ]) } + c ( p − (cid:27) . Then mimw( G ) ≤ h . Moreover, given a branch decomposition ( T i , δ i ) for G [ X i ] for each i , we can constructin O (1) time a branch decomposition ( T, δ ) for G with mimw G ( T, δ ) ≤ h . We use the following lemma as a base case for the proof of Theorem 2. (cid:73)
Lemma 9.
Let G be a (1 , ∆) -tree convex graph, for some ∆ ≥ . Let f (∆) = max ( $(cid:18) ∆2 (cid:19) % , − ) . Then mimw( G ) ≤ f (∆) . Moreover, we can construct in polynomial time a branch decomposition ( T, δ ) for G with mimw G ( T, δ ) ≤ f (∆) . Proof.
Let G = ( A, B, E ) be a (1 , ∆)-tree convex graph and let T be a (1 , ∆)-tree with V ( T ) = A andsuch that, for each v ∈ B , N G ( v ) forms a subtree of T . By Lemma 7, we can compute T in polynomialtime. Without loss of generality, ∆( T ) = ∆. Let u be the ∆-vertex in T and let C , . . . , C ∆ be the Jiang et al. [38] proved that
Weighted Feedback Vertex Set is polynomial-time solvable for triad convex graphs ifthe associated triad support is given as input. They observed that an associated tree support can be constructed inlinear time, but this does not imply that a triad support can be obtained. Lemma 7 shows that indeed a triad supportcan be obtained in polynomial time and need not be provided on input. . Brettell, A. Munaro, and D. Paulusma 9 components of T − u . To show the bound on the mim-width, in view of Lemma 8, we build a partition( X , . . . , X ∆ ) of V ( G ) as follows. Let B = { b ∈ B : N G ( b ) ⊆ { u }} be the set of vertices in B with degree 0or having u as the only neighbour. For each i ∈ { , . . . , ∆ } , let B i = { b ∈ B \ B : N G ( b ) ⊆ C i ∪ { u }} .Finally, let B = B \ ( ∪ ∆ i =1 B i ∪ B ). Since N G ( v ) forms a subtree of T , for each v ∈ B , it is not difficultto see that ( B ∪ B , B , . . . , B ∆ , B ) is a partition of B . We then let X = C ∪ B ∪ { u } ∪ B ∪ B and,for each i ∈ { , . . . , ∆ } , X i = C i ∪ B i .By Lemma 8, it suffices to show that cutmim G ( X i , X j ) ≤ i, j ∈ { , . . . , ∆ } and thatmimw( G [ X i ]) ≤ i ∈ { , . . . , ∆ } . The latter follows from the fact that G [ X i ] is convex for each i ∈ { , . . . , ∆ } and convex graphs have mim-width at most 1 [2]. Consider now the former. By construction,cutmim G ( X i , X j ) = 0 for all distinct i, j ∈ { , . . . , ∆ } . We finally show that cutmim G ( X , X j ) ≤ j ∈ { , . . . , ∆ } . Let M be a maximum induced matching in G [ X , X j ]. If | M | >
2, then there existtwo matching edges xy, x y ∈ M such that { x, x } ⊆ C j and { y, y } ⊆ B . Without loss of generality, d T ( x, u ) < d T ( x , u ). But each v ∈ B is adjacent to u and N G ( v ) forms a subtree of T . Therefore, y isadjacent to x as well, contradicting the fact that M is induced.The second assertion follows from Lemma 8 and the fact that we can construct in polynomial time abranch decomposition with mim-width at most 1 for each convex graph [2]. (cid:74) We are now ready to prove Theorem 2. (cid:73)
Theorem 2 (restated).
Let G be a ( t, ∆) -tree convex graph with t, ∆ ∈ N and t ≥ and ∆ ≥ . Let f ( t, ∆) = max ( $(cid:18) ∆2 (cid:19) % , − ) + 2 t − ∆ . Then mimw( G ) ≤ f ( t, ∆) . Moreover, we can construct in polynomial time a branch decomposition ( T, δ ) for G with mimw G ( T, δ ) ≤ f ( t, ∆) . Proof.
We proceed by induction on t . If t = 1, the result follows from Lemma 9. Therefore, supposethat t > G = ( A, B, E ) be a ( t, ∆)-tree convex graph. By Lemma 7, we can compute inpolynomial time a ( t, ∆)-tree T with V ( T ) = A and such that, for each v ∈ B , N G ( v ) forms a subtree of T .Consider an edge uv ∈ E ( T ) such that T − uv is the disjoint union of a ( t , ∆)-tree T containing u and a( t , ∆)-tree T containing v , where max { t , t } < t and t , t ≥
1. Clearly such an edge can be found inlinear time. For i ∈ { , } , let V ( T i ) = A i . Clearly, A = A ∪ A . We now partition B into two classes asfollows. The set B contains all vertices in B with at least one neighbour in A , and B = B \ B . Inview of Lemma 8, we then consider the partition ( A ∪ B , A ∪ B ) of V ( G ). For i ∈ { , } , G [ A i ∪ B i ] isa ( t i , ∆)-tree convex graph with t i < t and so, by the induction hypothesis,mimw( G [ A i ∪ B i ]) ≤ max ( $(cid:18) ∆2 (cid:19) % , − ) + 2 t − ∆ . We now claim that cutmim G ( A ∪ B , A ∪ B ) ≤ ∆( t − M be a maximum induced matchingin G [ A ∪ B , A ∪ B ]. Since no vertex in B has a neighbour in A , all edges in M have one endpoint in B and the other in A . We now consider the ( t , ∆)-tree T as a tree rooted at v , so that the nodes of T inherit a corresponding ancestor/descendant relation. Since T has maximum degree at most ∆ andcontains at most t vertices of degree at least 3, it has at most ∆ t ≤ ∆( t −
1) leaves. Suppose, to thecontrary, that | M | > ∆( t − xy, x y ∈ M with { y, y } ⊆ A and suchthat y is a descendant of y . Indeed, for each leaf z of T , consider the unique z, v -path in T . Thereare at most ∆( t −
1) such paths and each vertex of T is contained in one of them. By the pigeonholeprinciple, there exist two matching edges xy, x y ∈ M , with { y, y } ⊆ A , such that y and y belong tothe same path; without loss of generality, y is then a descendant of y , as claimed. Since N G ( x ) induces asubtree of T , the definition of ( A ∪ B , A ∪ B ) implies that N G ( x ) ∩ V ( T ) contains v and induces a subtree of T . But then this subtree contains y and so x is adjacent to y as well, contradicting the factthat M is induced.Combining the previous paragraphs and Lemma 8, we then obtain thatmimw( G ) ≤ max ( ∆( t − , max ( $(cid:18) ∆2 (cid:19) % , − ) + 2 t − ∆ + ∆( t − ) = max ( $(cid:18) ∆2 (cid:19) % , − ) + 2 t − ∆ + ∆( t − ≤ max ( $(cid:18) ∆2 (cid:19) % , − ) + 2 t − ∆ , as claimed.A branch decomposition for G can be computed recursively with the aid of Lemma 8 and Lemma 9. (cid:74) In this section we prove Theorem 3. We need the following procedure for constructing star convex andcomb convex graphs. (cid:73)
Lemma 10 (see, e.g., Wang et al. [62]) . Let G = ( A, B, E ) be a bipartite graph and let G be the bipartitegraph obtained from G by adding k new vertices, each complete to B . If k = 1 , then G is star convex,and if k = | A | , then G is comb convex. The following lemma follows from the fact that vertex deletion does not increase the mim-width of agraph [60]. (cid:73)
Lemma 11.
Let G = ( A, B, E ) be a bipartite graph and let G be the bipartite graph obtained from G by adding a new vertex complete to B . Then mimw( G ) ≥ mimw( G ) . We are now ready to prove Theorem 3. (cid:73)
Theorem 3 (restated).
The classes of star convex graphs and comb convex graphs have unboundedmim-width.
Proof.
We show that, for every integer ‘ , there exist star convex graphs and comb convex graphs withmim-width larger than ‘ . Therefore, let ‘ ∈ N . There exists a bipartite graph G = ( A, B, E ) such thatmimw( G ) > ‘ (see, e.g., [10]). Let G be the star convex graph obtained as in Lemma 10. By Lemma 11,mimw( G ) ≥ mimw( G ) > ‘ . Let now G be the comb convex graph obtained as in Lemma 10. Byrepeated applications of Lemma 11, mimw( G ) ≥ mimw( G ) > ‘ . (cid:74) We determined the underlying reason for several complexity dichotomies in the literature related togeneralized convex graphs by showing (un)boundedness of mim-width. We also determined new complexitydichotomies in this way. It would be interesting to obtain such dichotomies for other graph problems thatare solvable in polynomial time for graph classes whose mim-width is bounded and quickly computable.For example, what is the complexity of
List k -Colouring ( k ≥
3) for star convex and comb convexgraphs? We note that determining the complexity of
List -Colouring for chordal bipartite graphs isalso still open (see [25, 32, 53]), but List -Colouring is NP -complete even for P -free chordal bipartitegraphs [32], and thus also for P -free tree convex graphs. EFERENCES 11
The notion of generalized convex graphs plays a role in other settings as well. For example, Chenet al. [17] consider the problem
Subset Interconnection Design , which is, in our terminology, theproblem of deciding if a bipartite graph belongs to a class of H -convex graphs. This problem and its variantshave several applications, for example in the design of scalable overlay networks and vacuum systems (see[17]). Moreover, generalized convex graphs play an important role in combinatorial auctions [20, 28, 57]and fair allocation of indivisible goods [6, 33]. It would therefore be very interesting to research whetherthe problems in these settings are solvable for graph classes whose mim-width is bounded and quicklycomputable. We leave this for future research. References Bogdan Alecu, Aistis Atminas, and Vadim V. Lozin. Graph functionality.
Proc. WG 2019, LNCS ,11789:135–147, 2019. Rémy Belmonte and Martin Vatshelle. Graph classes with structured neighborhoods and algorithmicapplications.
Theoretical Computer Science , 511:54–65, 2013. Benjamin Bergougnoux and Mamadou Moustapha Kanté. More applications of the d-neighborequivalence: Connectivity and acyclicity constraints.
Proc. ESA 2019, LIPIcs , 144:17:1–17:14, 2019. Benjamin Bergougnoux, Charis Papadopoulos, and Jan Arne Telle. Node multiway cut and subsetfeedback vertex set on graphs of bounded mim-width.
Proc. WG 2020, LNCS , to appear, 2020. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractableFO model checking.
Proc. FOCS 2020 , to appear, 2020. Sylvain Bouveret, Katarína Cechlárová, Edith Elkind, Ayumi Igarashi, and Dominik Peters. Fairdivision of a graph.
Proc. IJCAI 2017 , pages 135–141, 2017. Andreas Brandstädt and Vadim V. Lozin. On the linear structure and clique-width of bipartitepermutation graphs.
Ars Combinatoria , 67, 2003. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad.
Graph Classes: A Survey . SIAMMonographs on Discrete Mathematics and Applications, Philadelphia, PA, 1999. Johann Brault-Baron, Florent Capelli, and Stefan Mengel. Understanding model counting for beta-acyclic cnf-formulas.
Proc. STACS 2015, LIPIcs , 30:143–156, 2015. Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, and Daniël Paulusma. Bounding themim-width of hereditary graph classes.
CoRR , abs/2004.05018, 2020. Nick Brettell, Jake Horsfield, and Daniël Paulusma. Colouring ( sP + P )-free graphs: a mim-widthperspective. CoRR , abs/2004.05022, 2020. Nick Brettell, Andrea Munaro, and Daniël Paulusma. List k -colouring P t -free graphs with no induced1-subdivision of K ,s : a mim-width perspective. CoRR , abs/2008.01590, 2020. Kevin Buchin, Marc van Kreveld, Henk Meijer, Bettina Speckmann, and Kevin Verbeek. On planarsupports for hypergraphs.
Journal of Graph Algorithms and Applications , 15:533–549, 2011. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Boolean-width of graphs.
TheoreticalComputer Science , 412:5187–5204, 2011. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Fast dynamic programming for locallycheckable vertex subset and vertex partitioning problems.
Theoretical Computer Science , 511:66–76,2013. Hao Chen, Zihan Lei, Tian Liu, Ziyang Tang, Chaoyi Wang, and Ke Xu. Complexity of domination,hamiltonicity and treewidth for tree convex bipartite graphs.
Journal of Combinatorial Optimization ,32:95–110, 2016. Jiehua Chen, Christian Komusiewicz, Rolf Niedermeier, Manuel Sorge, Ondřej Suchý, and MathiasWeller. Polynomial-time data reduction for the Subset Interconnection Design problem.
SIAM Journalon Discrete Mathematics , 29:1–25, 2015. Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. List 3-coloring P t -free graphs with no induced1-subdivision of K ,s . Discrete Mathematics , to appear. Maria Chudnovsky, Jason King, Michał Pilipczuk, Paweł Rzążewski, and Sophie Spirkl. Finding large H -colorable subgraphs in hereditary graph classes. CoRR , abs/2004.09425, 2020. Vincent Conitzer, Jonathan Derryberry, and Tuomas Sandholm. Combinatorial auctions with struc-tured item graphs.
Proc. AAAI 2014 , pages 212–218, 2004. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs.
Discrete AppliedMathematics , 101:77–114, 2000. Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, and Daniël Paulusma. List coloring in theabsence of a linear forest.
Algorithmica , 71:21–35, 2015. Konrad K. Dabrowski, Matthew Johnson, and Daniël Paulusma. Clique-width for hereditary graphclasses.
London Mathematical Society Lecture Note Series , 456:1–56, 2019. Peter Damaschke, Haiko Müller, and Dieter Kratsch. Domination in convex and chordal bipartitegraphs.
Information Processing Letters , 36:231–236, 1990. Josep Díaz, Öznur Yasar Diner, Maria J. Serna, and Oriol Serra. On list k -coloring convex bipartitegraphs. Proc. CTW 2020, AIRO Springer Series , to appear. Esther Galby, Andrea Munaro, and Bernard Ries. Semitotal domination: New hardness results anda polynomial-time algorithm for graphs of bounded mim-width.
Theoretical Computer Science , 814:28–48, 2020. Fred Glover. Maximum matching in a convex bipartite graph.
Naval Research Logistics Quarterly , 14:313–316, 1967. Georg Gottlob and Gianluigi Greco. On the complexity of combinatorial auctions: Structured itemgraphs and hypertree decomposition.
Proc. EC 2007 , page 152–161, 2007. Frank Gurski. The behavior of clique-width under graph operations and graph transformations.
Theoryof Computing Systems , 60:346–376, 2017. Petr Hliněný, Sang-il Oum, Detlef Seese, and Georg Gottlob. Width parameters beyond tree-widthand their applications.
The Computer Journal , 51:326–362, 2008. Chính T. Hoàng, Marcin Kamiński, Vadim V. Lozin, Joe Sawada, and Xiao Shu. Deciding k -colorabilityof P -free graphs in polynomial time. Algorithmica , 57:74–81, 2010. Shenwei Huang, Matthew Johnson, and Daniël Paulusma. Narrowing the complexity gap for colouring( C s , P t )-free graphs. Computer Journal , 58:3074–3088, 2015. Ayumi Igarashi and Dominik Peters. Pareto-optimal allocation of indivisible goods with connectivityconstraints.
Proc. AAAI 2019 , pages 2045–2052, 2019. Lars Jaffke, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. Mim-width III. Graph powersand generalized distance domination problems.
Theoretical Computer Science , 796:216–236, 2019. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width I. Induced path problems.
DiscreteApplied Mathematics , 278:153–168, 2020. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width II. The Feedback Vertex Set problem.
Algorithmica , 82:118–145, 2020. Wei Jiang, Tian Liu, and Ke Xu. Tractable feedback vertex sets in restricted bipartite graphs.
Proc.COCOA 2011, LNCS , 6831:424–434, 2011. Wei Jiang, Tian Liu, Chaoyi Wang, and Ke Xu. Feedback vertex sets on restricted bipartite graphs.
Theoretical Computer Science , 507:41–51, 2013. Öjvind Johansson. Clique-decomposition, NLC-decomposition, and modular decomposition - relation-ships and results for random graphs.
Congressus Numerantium , 132:39–60, 1998. W. Lipski Jr. and F. P. Preparata. Efficient algorithms for finding maximum matchings in convexbipartite graphs and related problems.
Acta Informatica , 15:329–346, 1981.
EFERENCES 13 Marcin Kamiński, Vadim V. Lozin, and Martin Milanič. Recent developments on graphs of boundedclique-width.
Discrete Applied Mathematics , 157:2747–2761, 2009. Dong Yeap Kang, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. A width parameteruseful for chordal and co-comparability graphs.
Theoretical Computer Science , 704:1–17, 2017. O-joung Kwon. Personal communication. 2020. Y. Daniel Liang and Norbert Blum. Circular convex bipartite graphs: Maximum matching andhamiltonian circuits.
Information Processing Letters , 56:215–219, 1995. Tian Liu. Restricted bipartite graphs: Comparison and hardness results.
Proc. AAIM 2014, LNCS ,8546:241–252, 2014. Tian Liu, Min Lu, Zhao Lu, and Ke Xu. Circular convex bipartite graphs: Feedback vertex sets.
Theoretical Computer Science , 556:55–62, 2014. Tian Liu, Zhao Lu, and Ke Xu. Tractable connected domination for restricted bipartite graphs.
Journal of Combinatorial Optimization , 29:247–256, 2015. Min Lu, Tian Liu, and Ke Xu. Independent domination: Reductions from circular- and triad-convexbipartite graphs to convex bipartite graphs.
Proc. FAW-AAIM 2013, LNCS , 7924:142–152, 2013. Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width.
Journal ofCombinatorial Theory, Series B , 96:514–528, 2006. B. S. Panda and Juhi Chaudhary. Dominating induced matching in some subclasses of bipartitegraphs.
Proc. CALDAM 2019, LNCS , 11394:138–149, 2019. B. S. Panda, Arti Pandey, Juhi Chaudhary, Piyush Dane, and Manav Kashyap. Maximum weightinduced matching in some subclasses of bipartite graphs.
Journal of Combinatorial Optimization , toappear. Arti Pandey and B.S. Panda. Domination in some subclasses of bipartite graphs.
Discrete AppliedMathematics , 252:51–66, 2019. Daniël Paulusma. Open problems on graph coloring for special graph classes.
Proc. WG 2015, LNCS ,9224:16–30, 2015. Michaël Rao. Clique-width of graphs defined by one-vertex extensions.
Discrete Mathematics , 308:6157–6165, 2008. Neil Robertson and Paul D. Seymour. Graph minors. X. Obstructions to tree-decomposition.
Journalof Combinatorial Theory, Series B , 52:153–190, 1991. Sigve Hortemo Sæther and Martin Vatshelle. Hardness of computing width parameters based onbranch decompositions over the vertex set.
Theoretical Computer Science , 615:120–125, 2016. Tuomas Sandholm and Subhash Suri. BOB: Improved winner determination in combinatorial auctionsand generalizations.
Artificial Intelligence , 145:33–58, 2003. Yu Song, Tian Liu, and Ke Xu. Independent domination on tree convex bipartite graphs.
Proc.FAW-AAIM 2012, LNCS , 7285:129–138, 2012. Jan Arne Telle and Andrzej Proskurowski. Algorithms for vertex partitioning problems on partial k -trees. SIAM Journal on Discrete Mathematics , 10:529–550, 1997. Martin Vatshelle.
New Width Parameters of Graphs . PhD thesis, University of Bergen, 2012. Chaoyi Wang, Tian Liu, Wei Jiang, and Ke Xu. Feedback vertex sets on tree convex bipartite graphs.
Proc. COCOA 2012, LNCS , 7402:95–102, 2012. Chaoyi Wang, Hao Chen, Zihan Lei, Ziyang Tang, Tian Liu, and Ke Xu. Tree convex bipartite graphs:
N P -complete domination, hamiltonicity and treewidth.