Solving Problems on Generalized Convex Graphs via Mim-Width

A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of $H$. Many $\mathsf{NP}$-complete problems, including problems such as Dominating Set, Feedback Vertex Set, Induced Matching and List $k$-Colouring, become polynomial-time solvable for ${\mathcal H}$-convex graphs when ${\mathcal H}$ is the set of paths. In this case, the class of ${\mathcal H}$-convex graphs is known as the class of convex graphs. The underlying reason is that the class of convex graphs has bounded mim-width. We extend the latter result to families of ${\mathcal H}$-convex graphs where (i) ${\mathcal H}$ is the set of cycles, or (ii) ${\mathcal H}$ 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 known in the literature. To complement result (ii), we show that the mim-width of ${\mathcal H}$-convex graphs is unbounded if ${\mathcal H}$ is the set of trees with arbitrarily large maximum degree or an 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. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of ${\cal H}$-convex graphs.


Introduction
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 to seek the underlying reason why some graph problems are easier on certain classes of input than others. A general 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 for many problems simultaneously. This requires some notion of an "optimal" vertex decomposition, which depends on the type of vertex decomposition used and which may relate to the minimum number of sets or 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 some constant c. Boundedness of width is often the underlying reason why a graph-class-specific algorithm runs efficiently: in such a case, the proof that the algorithm is efficient for some special graph class reduces to a proof showing that the width of the class is bounded by some constant. We will see illustratations of this 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 width parameter 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 other, then p and q are equivalent. If p is more powerful than q, then the class of graphs for which p is bounded 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 fewer problems 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 parameters continue to be defined, such as graph functionality [1] in 2019 and twin-width [5] in 2020. The well-known parameters boolean-width, clique-width, module-width, NLC-width and rank-width are all equivalent to each other [14,39,49,54]. They are more powerful than the equivalent parameters branch-width and treewidth [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 be tractable on graph classes of bounded width. Proving boundedness of width of some graph class often immediately tells us that many problems are tractable for that class without the need for constructing algorithms for each problem. Hence, it is natural to ask which structural properties of a graph class ensure constant-bounded width.
Our Focus. There are large families of natural graph classes for which the (un)boundedness of width is not known for many width parameters. We will focus on a relatively new width parameter, namely the 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 of problems including k-Colouring and its generalization List k-Colouring (the algorithms are given in [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 such classes are obtained by reducing to the class of convex graphs. We show that our approach via mim-width simplifies the analysis, unifies the sporadic approaches and explains the reductions to convex graphs.

Mim-width
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 the vertex set of a graph G. Then G[A, A] denotes the bipartite subgraph of G induced by the edges with one endpoint in A and the other in A. Motivated by algorithmic applications, Vatshelle [60] introduced the notion of maximum induced matching width, also called mim-width. Mim-width measures the extent to which it is possible to decompose a graph G along certain vertex partitions (A, A) such that the size of a maximum induced matching in G[A, A] is small. The kind of vertex partitions permitted stem from classical branch decompositions. A branch decomposition for a graph G is a pair (T, δ), where T is a subcubic tree and δ is a bijection from V (G) to the leaves of T . Every edge e ∈ E(T ) partitions the leaves of T into two classes, L e and L e , depending on which component of T − e they belong to. Hence, e induces a 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 a maximum induced matching in G[A e , A e ]. Then the mim-width mimw G (T, δ) of (T, δ) is the maximum value of cutmim G (A e , A e ) over all edges e ∈ E(T ). The mim-width mimw(G) of G is the minimum value of 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 in polynomial 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 decomposition whose mim-width is bounded by a constant. If this is possible, the mim-width of G is said to be quickly computable. One can then try to develop a polynomial-time algorithm for the graph problem under consideration via dynamic programming over the computed branch decomposition. We briefly discuss a number 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 Locally Checkable Vertex Subset and Vertex Partitioning (LC-VSVP) problems, first defined in [59], are polynomialtime solvable on graph classes whose mim-width is bounded and quickly computable. Examples of such problems are (Dominating) Induced Matching, (Total) Dominating Set, Independent Dominating 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 for graph classes whose mim-width is bounded and quickly computable. Jaffke et al. [35,36] [3] proved that Connected Dominating Set, Node Weighted Steiner Tree and Maximum Induced Tree are all polynomial-time solvable on graph classes whose mim-width is bounded and quickly computable. Galby et al. [26] proved the same for Semitotal Dominating Set, whereas Chudnovsky et al. [19] observed this for Max Partial H-Colouring, a problem that generalizes, amongst others, Odd Cycle Transversal.

Convex Graphs and Generalizations
A bipartite graph G = (A, B, E) is convex if there exists a path P with V (P ) = A such that the neighbours in A of each b ∈ B induce a connected subpath of P . Convex graphs generalize bipartite permutation graphs (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 early paper 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,  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 give another 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-time solvable for any class of graphs whose mim-width is bounded and quickly computable [43] (see [11]), a polynomial-time algorithm also directly follows from the result of [2]. In the remainder of our paper, we consider superclasses of convex graphs and research to what extent mim-width can play a role in obtaining polynomial-time algorithms for problems on these classes.
Let H be a family of graphs.
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 a backbone consisting of one vertex is a star. A comb is a caterpillar such that every backbone vertex has exactly 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 of subdivisions. 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, 3)-tree. Now, if H consists of all cycles, all trees, all stars, all triads, all combs or all (t, ∆)-trees, then we obtain the class of circular convex graphs, tree convex graphs, star convex graphs, triad convex graphs, comb convex graphs 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,∆ be the class of (t, ∆)-tree convex graphs. For fixed t or ∆, we have increasing sequences C t,0 ⊆ C t,1 ⊆ · · · and C 0,∆ ⊆ C 1,∆ ⊆ · · · . For t ∈ N, the class of (t, ∞)-tree convex graphs is ∆∈N C t,∆ , denoted by C t,∞ . Similarly, for ∆ ∈ N, the class of (∞, ∆)-tree convex graphs is 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 t,∆∈N C t,∆ , which coincides with the class of tree convex graphs. Notice that the class of convex graphs coincides with C t,2 , for any t ∈ N ∪ {∞}, and with C 0,∆ , for any ∆ ∈ N ∪ {∞}. The class of star convex graphs coincides with C 1,∞ . Moreover, each triad convex graph belongs to C 1,3 and each comb convex graph belongs to C ∞,3 . A bipartite graph is chordal bipartite if every 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 other relationships, which directly follow from the definitions.
Brault-Baron et al. [9] proved that chordal bipartite graphs have unbounded mim-width and so the result 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 known algorithmic results for these classes.

Known Results
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 convex graphs and triad convex graphs, but NP-complete for star convex and comb convex graphs. Pandey and Panda [52] proved that Dominating Set is polynomial-time solvable for circular convex, triad convex and (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] showed that Dominating Set, Connected Dominating Set and Total Dominating Set are NP-complete The inclusion relations between the classes mentioned in our paper. A line from a lower-level class to a higher one indicates that the first class is contained in the second. The dotted line separates the classes of bounded 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 circular convex and triad convex graphs. The latter result was shown already by Song et al. [58] who used a dynamic 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 solvable for (t, ∆)-tree convex graphs for every t ≥ 1 and ∆ ≥ 3. They also showed in [58] that Independent Dominating Set is NP-complete for star convex graphs and for comb convex graphs. Hence, they obtained a dichotomy: Independent Dominating Set is polynomial-time solvable for (t, ∆)-tree convex graphs for every t ≥ 1 and ∆ ≥ 3 but NP-complete for (∞, 3)-tree convex graphs and for (1, ∞)-tree convex graphs.
The same dichotomy (explicitly formulated in [61]) holds for Feedback Vertex Set and is obtained similarly. Namely, Jiang et al. [37] proved that this problem is polynomial-time solvable for triad convex graphs and mentioned that their algorithm can be generalized to (t, ∆)-tree convex graphs for every t ≥ 1 and ∆ ≥ 3. Jiang et al. [38] proved that Feedback Vertex Set is NP-complete for star convex and comb convex graphs. In addition, Liu et al. [46] proved that Feedback Vertex Set is polynomial-time solvable for circular convex graphs, whereas Jiang et al. [38] proved that the Weighted Feedback Vertex 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 and subclasses 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] proved that Dominating Induced Matching is not only polynomial-time solvable on circular convex and triad convex graphs, but also on star convex graphs. Nevertheless, we notice a common pattern: many dominating set, induced matching and graph transversal type of problems are polynomial-time solvable for (t, ∆)-tree convex graphs, for every t ≥ 1 and ∆ ≥ 3, and NP-complete for comb convex graphs, and thus for (∞, 3)-tree convex graphs, and star convex graphs, or equivalently, (1, ∞)-tree convex graphs. Moreover, essentially all the polynomial-time algorithms reduce the input to a convex graph.

Our Results
As mentioned, our goal is to simplify the analysis, unify the approaches in Section 1.3 and explain the reductions to convex graphs, using mim-width.
Structural Results. We prove the following results explaining the dotted line in Figure 2. The first two results generalize that of Belmonte and Vatshelle [2], as convex graphs form a common subclass of circular convex graphs and (1, 3)-tree convex graphs. The third result gives two new reasons why tree convex graphs (or equivalently (∞, ∞)-tree convex graphs) have unbounded mim-width. Theorem 1. Let G be a circular convex graph. Then mimw(G) ≤ 2. Moreover, we can construct in polynomial time a branch decomposition (T, δ) for G with mimw G (T, δ) ≤ 2.

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 (∞, 3)-tree convex graphs). Algorithmic Consequences. As explained in Section 1.3, the following six problems were shown to be NP-complete for both star convex graphs and comb convex graphs, and thus for (1, ∞)-tree convex graphs and (∞, 3)-tree convex graphs: Feedback Vertex Set [38]; Dominating Set, Connected Dominating Set, Total Dominating Set [16]; Independent Dominating Set [58]; Induced Matching [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 quickly computable [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 Feedback Vertex Set and is thus NP-complete for star convex graphs and comb convex graphs. Combining these results with Corollary 4 yields the following complexity dichotomy. 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 1 and 2, combined with the result of [15], imply that this problem is also polynomial-time solvable on circular convex graphs and (t, ∆)-tree convex graphs for every t ≥ 1 and ∆ ≥ 3. On another note, Theorems 1 and 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 ≥ 1, ∆ ≥ 3).
It remains to prove Theorems 1-3, which we do in Sections 3-5, respectively.

Preliminaries
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.  For a graph H, a graph G is H-free if G has no induced subgraph isomorphic to H. The path on n vertices is denoted by P n . A graph is r-partite, for r ≥ 2, if its vertex set admits a partition into r classes such that every edge has its endpoints in different classes. A 2-partite graph is also called bipartite. A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to 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.

The Proof of Theorem 1
In this section we prove Theorem 1. We need the following known lemma on recognizing circular convex graphs.

. Circular convex graphs can be recognized and a cycle support computed, if it exists, in polynomial time.
For an integer ≥ 1, an -caterpillar is a subcubic tree T on 2 vertices with V (T ) = {s 1 , . . . , s , t 1 Note that we label the leaves of an -caterpillar t 1 , t 2 , . . . , t , in this order. Given a total ordering ≺ of length , we say that (T, δ) is obtained from ≺ 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.
Proof. Let G = (A, B, E) be a circular convex graph with a circular ordering on A. By Lemma 6, we construct in polynomial time such an ordering a 1 , . . . , a n , where n = |A| (see Figure 1). Let B 1 = {b ∈ B : a n ∈ N (b)} and B 2 = B \ B 1 . We obtain a total ordering ≺ on V (G) by extending the ordering a 1 , . . . , a n as follows. Each b ∈ B 1 is inserted after a n , breaking ties arbitrarily. Each b ∈ B 2 is inserted immediately after the largest element of A it is adjacent to (hence immediately after some a i with 1 ≤ i < n), breaking ties arbitrarily.
Let T be the |V (G)|-caterpillar obtained from ≺. We show that mimw G (T, δ) ≤ 2. Let e be an edge of T and let M be a maximum induced matching of G[A e , A e ], where each vertex in A e is larger than any vertex in A e . Observe first that at most one edge of M has one endpoint in B 2 . Indeed, suppose there exist two edges xy, x y ∈ M , each with one endpoint in B 2 , say without loss of generality {y, y } ⊆ B 2 . Since each vertex in B 2 is adjacent only to smaller vertices, {y, y } ⊆ A e and {x, x } ⊆ A e . Without loss 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 1 and, if exactly two such edges are in M , then no edge with an endpoint in B 2 is. Suppose, to the contrary, that three edges of M have one endpoint in B 1 and let u 1 , u 2 , u 3 be these endpoints. Since N (u 1 ), N (u 2 ) and N (u 3 ) 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 is induced. Suppose finally that exactly two edges u 1 v 1 , u 2 v 2 ∈ M have one endpoint in B 1 . We may assume {u 1 , u 2 } ⊆ A e , {v 1 , v 2 } ⊆ A e and u 1 ∈ B 1 . Then u 2 ∈ B 1 and so {v 1 , v 2 } ⊆ A. Now if there is some edge u 3 v 3 ∈ M such that u 3 ∈ B 2 , then u 3 ∈ A e . Recall that N (u 1 ) and N (u 2 ) are intervals of the circular ordering on A both containing a n . Since M is induced, for each i, j ∈ {1, 2}, we have that This implies that one of v 1 and v 2 is larger than v 3 in ≺ and so it is contained in N (u 3 ), contradicting the fact that M is induced. This concludes the proof.

4
The Proof of Theorem 2 In this section we prove Theorem 2. We need the following lemma on recognizing (t, ∆)-tree convex graphs 1 . = O(|A| t ) subsets A ⊆ A of size 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, ∆)-tree convex graph, the algorithm returns a (t, ∆)-tree support for H.
The proof of Theorem 2 heavily relies on the following general result for mim-width. We use the following lemma as a base case for the proof of Theorem 2.  B , B 2 , . . . , B ∆ , B ) is a partition of B. We then let By Lemma 8, it suffices to show that cutmim G (X i , X j ) ≤ 2 for all distinct i, j ∈ {1, . . . , ∆} and that mimw(G[X i ]) ≤ 1 for each i ∈ {1, . . . , ∆}. The latter follows from the fact that G[X i ] is convex for each i ∈ {1, . . . , ∆} 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 ∈ {2, . . . , ∆}. We finally show that cutmim G (X 1 , X j ) ≤ 2 for each j ∈ {2, . . . , ∆}. Let M be a maximum induced matching in G[X 1 , X j ]. If |M | > 2, then there exist two matching edges xy, x y ∈ M such that {x, x } ⊆ C j and {y, y } ⊆ B . Without loss of generality, u). But each v ∈ B is adjacent to u and N G (v) forms a subtree of T . Therefore, y is adjacent 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 a branch decomposition with mim-width at most 1 for each convex graph [2].
We are now ready to prove Theorem 2.

Proof.
We proceed by induction on t. If t = 1, the result follows from Lemma 9. Therefore, suppose that t > 1 and let G = (A, B, E) be a (t, ∆)-tree convex graph. By Lemma 7, we can compute in polynomial 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 1 , ∆)-tree T 1 containing u and a (t 2 , ∆)-tree T 2 containing v, where max{t 1 , t 2 } < t and t 1 , t 2 ≥ 1. Clearly such an edge can be found in linear time. For i ∈ {1, 2}, let V (T i ) = A i . Clearly, A = A 1 ∪ A 2 . We now partition B into two classes as follows. The set B 1 contains all vertices in B with at least one neighbour in A 1 , and B 2 = B \ B 1 . In view of Lemma 8, we then consider the partition ( is a (t i , ∆)-tree convex graph with t i < t and so, by the induction hypothesis, We now claim that cutmim G ( . Since no vertex in B 2 has a neighbour in A 1 , all edges in M have one endpoint in B 1 and the other in A 2 . We now consider the (t 2 , ∆)-tree T 2 as a tree rooted at v, so that the nodes of T 2 inherit a corresponding ancestor/descendant relation. Since T 2 has maximum degree at most ∆ and contains at most t 2 vertices of degree at least 3, it has at most ∆t 2 ≤ ∆(t − 1) leaves. Suppose, to the contrary, that |M | > ∆(t − 1). We first claim that there exist xy, x y ∈ M with {y, y } ⊆ A 2 and such that y is a descendant of y. Indeed, for each leaf z of T 2 , consider the unique z, v-path in T 2 . There are at most ∆(t − 1) such paths and each vertex of T 2 is contained in one of them. By the pigeonhole principle, there exist two matching edges xy, x y ∈ M , with {y, y } ⊆ A 2 , such that y and y belong to the same path; without loss of generality, y is then a descendant of y, as claimed. Since N G (x ) induces a subtree of T , the definition of (A 1 ∪ B 1 , A 2 ∪ B 2 ) implies that N G (x ) ∩ V (T 2 ) contains v and induces a subtree of T 2 . But then this subtree contains y and so x is adjacent to y as well, contradicting the fact that M is induced.
Combining the previous paragraphs and Lemma 8, we then obtain that as claimed. A branch decomposition for G can be computed recursively with the aid of Lemma 8 and Lemma 9.

The Proof of Theorem 3
In this section we prove Theorem 3. We need the following procedure for constructing star convex and comb convex graphs.
Lemma 10 (see, e.g., Wang et al. [62]). Let G = (A, B, E) be a bipartite graph and let G be the bipartite graph 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 a graph [60]. We are now ready to prove Theorem 3.

Theorem 3 (restated). The classes of star convex graphs and comb convex graphs have unbounded mim-width.
Proof. We show that, for every integer , there exist star convex graphs and comb convex graphs with mim-width larger than . Therefore, let ∈ N. There exists a bipartite graph G = (A, B, E) such that mimw(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. By repeated applications of Lemma 11, mimw(G ) ≥ mimw(G) > .

Conclusions
We determined the underlying reason for several complexity dichotomies in the literature related to generalized convex graphs by showing (un)boundedness of mim-width. We also determined new complexity dichotomies in this way. It would be interesting to obtain such dichotomies for other graph problems that are 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 convex graphs? We note that determining the complexity of List 3-Colouring for chordal bipartite graphs is also still open (see [25,32,53]), but List 4-Colouring is NP-complete even for P 8 -free chordal bipartite graphs [32], and thus also for P 8 -free tree convex graphs.
The notion of generalized convex graphs plays a role in other settings as well. For example, Chen et al. [17] consider the problem Subset Interconnection Design, which is, in our terminology, the problem of deciding if a bipartite graph belongs to a class of H-convex graphs. This problem and its variants have 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 whether the problems in these settings are solvable for graph classes whose mim-width is bounded and quickly computable. We leave this for future research.