(α, β)-Modules in Graphs
aa r X i v : . [ c s . D M ] J a n ( α, β )-Modules in Graphs ⋆ Michel Habib , Lalla Mouatadid , Eric Sopena , and Mengchuan Zou IRIF, UMR 8243 CNRS & Paris University, Paris, France Department of Computer Science, University of Toronto, Toronto, ON, Canada Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France
Abstract.
Modular Decomposition focuses on repeatedly identifying amodule M (a collection of vertices that shares exactly the same neigh-bourhood outside of M ) and collapsing it into a single vertex. This notionof exactitude of neighbourhood is very strict, especially when dealing withreal world graphs.We study new ways to relax this exactitude condition. However, gener-alizing modular decomposition is far from obvious. Most of the previousproposals lose algebraic properties of modules and thus most of the nicealgorithmic consequences.We introduce the notion of an ( α, β )-module , a relaxation that allowsa bounded number of errors in each node and maintains some of thealgebraic structure. It leads to a new combinatorial decomposition withinteresting properties. Among the main results in this work, we show thatminimal ( α, β )-modules can be computed in polynomial time, and thatevery graph admits an ( α, β )-modular decomposition tree, thus general-izing Gallai’s Theorem (which corresponds to the case for α = β = 0).Unfortunately we give evidence that computing such a decompositiontree can be difficult. First introduced for undirected graphs by Gallai in [19] to analyze the structureof comparability graphs, modular decomposition has been used and defined inmany areas of discrete mathematics, including 2-structures, automata, partialorders, set systems, hypergraphs, clutters, matroids, boolean and submodularfunctions [13,14,17,21]. For a survey on modular decomposition, see [31] and forits algorithmic aspects [23]. Since they have been rediscovered in many fields,modules appear under various names in the literature, they have been calledintervals, externally related sets, autonomous sets, partitive sets, homogeneoussets, and clans. In most of the above examples the family of modules of a givengraph yields a kind of partitive family [6,8,9], and therefore leads to a uniquemodular decomposition tree that can be computed efficiently.Roughly speaking, elements of a module M behave exactly the same withrespect to elements outside of M . Thus a module can be contracted to a single ⋆ This work is supported by the ANR-France Project Hosigra (ANR-17-CE40-0022).Preliminary results of this work were presented in [22]. element without losing neighbourhood and connectivity information. This tech-nique has been used to solve many optimization problems and has led to a num-ber of elegant graph algorithms, see for instance [30]. Other direct applicationsof modular decomposition appear in areas such as computational protein-proteininteraction networks and graph drawing [18,35]. Recently, new applications haveappeared in the study of networks in social sciences [40], where a module is con-sidered as a regularity or a community that has to be detected and understood.Although it is well known that almost all graphs have no non-trivial mod-ules [32], some graphs that arise from real data seem to have many non-trivialmodules [34]. How can we explain such a phenomenon? It could be that the con-text in which this real data is generated has a clustering structure; but it couldalso be because we reach some known regularities as predicted by Szemer´edi’sRegularity Lemma [41]. In fact for every ǫ > n such that all undirected graphs with more than n verticesadmit an ǫ -regular partition of its vertices. Such a partition is a kind of an ap-proximate modular decomposition, and linear time algorithms for exact modulardecomposition are known [23]. Our results.
In this paper we introduce and study a new generalization of modulardecomposition by relaxing the strict neighbourhood condition of modules witha tolerance of some errors (missing or extra edges). In particular, we define an( α, β )-module to be a set M whose elements behave exactly the same with respectto elements outside of M , except that each outside element can have either atmost α missing edges or at most β extra edges connecting it to M . In otherwords, an ( α, β )-module M can be turned into a module by adding at most α edges, or deleting at most β edges, at each element outside M . In particular, werecover the standard modular decomposition when α = β = 0.This new combinatorial decomposition is not only theoretically interestingbut also can lead to practical applications. We first prove that every graphadmits an ( α, β )-modular decomposition tree which is a kind of generalizationof Gallai’s Theorem. But by no means such a tree is unique and we also giveevidence that finding such a tree could be NP-hard. On the algorithmic sidewe propose a polynomial algorithm to compute a covering of the vertex set byminimal ( α, β )-modules with a bounded overlap, in O ( m · n α + β +1 ) time. For thebipartite case, when we restrict ( α, β )-modules on one side of the bipartition, wecompletely compute all these ( α, β )-modules. In particular, we give an algorithmthat computes a covering of the vertices of a bipartite graph in O ( n α + β ( n + m ))time, using maximal ( α, β )-modules. This can be of great help for communitydetection in bipartite graphs. Organization of the paper.
Section 2 covers the necessary background on stan-dard modular decomposition, introduces ( α, β )-modules and illustrates variousapplications of ( α, β )-modular decomposition. Sections 3 covers structural prop-erties of ( α, β )-modules and the NP-hardness results. Section 4 contains all thealgorithmic results, in particular the computation of minimal ( α, β )-modules aswell as ( α, β )-primality testing. Section 5 covers the complete determination of ( α, β )-modules that lay one side of a bipartite graph. We conclude in Section 6with an alternate relaxation of modular decomposition. Let G = ( V ( G ) , E ( G )) be a graph on | V ( G ) | = n vertices and | E ( G ) | = m edges. For two adjacent vertices u, v ∈ V ( G ), uv denotes the edge in E ( G ) withendpoints u and v . All the graphs considered here are simple (no loops, nomultiple edges), finite and undirected. The complement of a graph G = ( V, E )is the graph G = ( V ( G ) , E ( G )) where uv ∈ E ( G ) if and only uv / ∈ E ( G ). Weoften refer to the sets of vertices and edges of G as V and E respectively, if G isclear from the context.For a set of vertices X ⊆ V , we denote by G ( X ) the induced subgraph of G generated by X . The set N ( v ) = { u : uv ∈ E } is the neighbourhood of v and theset N ( v ) = { u : uv / ∈ E } the non-neighbourhood of v . This notation can also beextended to sets of vertices: for a set X ⊆ V , we let N ( X ) = { x ∈ V \ X : ∃ y ∈ X and xy ∈ E ( G ) } , and N ( X ) = { x ∈ V \ X : ∀ y ∈ X, xy / ∈ E ( G ) } . Note here that N ( X ) is not the union of the sets N ( x ) for all x ∈ X , but theset of vertices outside from X that have a neighbour in X .Two vertices u and v are called false twins if N ( u ) = N ( v ), and true twins if N ( u ) ∪ { u } = N ( v ) ∪ { v } .A Moore family on a set X is a collection of subsets of X that contains X itself and is closed under intersection. Definition 1. A module of a graph G = ( V, E ) is a set of vertices M ⊆ V thatsatisfies ∀ x, y ∈ M, N ( x ) \ M = N ( y ) \ M. In other words, V \ M is partitioned into two parts A, B such that there isa complete bipartite subgraph between M and A , and no edges between M and B . Observe that we have A = N ( M ), and B = N ( M ). It is to easy to see thatevery two vertices within a module are either false twins or true twins.A single vertex { v : v ∈ V } is always a module, and so is the set V . Suchmodules are called trivial modules . A graph with only trivial modules is called a prime graph. A module is maximal if it is not contained in any other non-trivialmodule.A modular decomposition tree of a graph G is a tree T ( G ) that captures thedecomposition of G into modules. The leaves of T ( G ) represent the vertices of G ,the internal nodes of T ( G ) capture operations on modules, and are labelled par-allel, series, or prime . A parallel node captures the disjoint union of its children,whereas a series node captures the full connection of its children (i.e., connectevery vertex of its left child to every vertex of its right child). Parallel and series a • b • c • d • e • f • g • h • G Prime a • S e • P h • P c • f • g • b • d • Fig. 1.
A graph G (left) and its modular decomposition tree (right). Maximal modulesare red, series and parallel nodes are labelled in the tree as S and P respectively. nodes are often referred to as complete nodes. Fig. 1 illustrates a graph with itsmodular decomposition tree.By the Modular Decomposition Theorem [9,19], every graph admits a unique modular decomposition tree. Other combinatorial objects also admit unique de-composition trees, partitive families in particular.Two sets A and B overlap if A ∪ B = ∅ , A \ B = ∅ , and B \ A = ∅ . In a familyof subsets F of a ground set V , a set S ∈ F is strong if S does not overlap withany other set in F . We denote by ∆ the symmetric difference of two sets: A∆B = { a : a ∈ A \ B } ∪ { b : b ∈ B \ A } . Definition 2 ([9]).
A family of subsets F over a ground set V is partitive if(i) ∅ , V , and all singletons { x : x ∈ V } belong to F , and(ii) ∀ A, B ∈ F , if A ∩ B = ∅ then A ∪ B ∈ F , A ∩ B ∈ F , A \ B ∈ F , and A∆B ∈ F . Partitive families play a fundamental role in combinatorial decomposition [8,9].Every partitive family admits a unique decomposition tree with only completeand prime nodes. The strong elements of F form a tree ordered by the inclusionrelation [9].A complement reducible graph is a graph whose decomposition tree has noprime nodes, that is, the graph is totally decomposable into parallel and seriesnodes only. Complement reducible graphs are also known as cographs , and areexactly the P -free graphs [39]. A modular decomposition tree of a cograph isoften referred to as a cotree . Cographs have been widely studied, and manytypical N P -hard problems (colouring, independent set, etc.) become tractableon cographs [11].
Finding a non-trivial tractable generalization of modules is not an easy task.Indeed, when trying to do so, we are faced with two main difficulties.
The first one is to obtain a pseudo-generalization. Suppose for example thatwe change the definition of a module into: ∀ x, y ∈ M , N ∗ ( x ) \ M = N ∗ ( y ) \ M , where N ∗ ( x ) can mean something like “vertices at distance at most k ” or“vertices joined by an odd path”, etc. In many of these scenarios, it turns outthat the problem transforms itself into the computation of precisely the modulesof some auxiliary graph built from the original one. Some work in this directionavoiding this drawback can be found in [7].The second difficulty is N P -hardness. Consider the notion of roles defined insociology, where two vertices play the same role in a social network if they havethe same set of colours in their neighbourhood. In this scenario, if a colouringof the vertices is given, then one can compute these roles in polynomial time.Otherwise, the problem is indeed a colouring problem which is
N P -hard tocompute [16].In this work, we consider two variations of the notion of modules, both ofwhich trying to avoid these two difficulties. Some of these new modules arepolynomial to compute, and we believe they are worth studying further. Wefocus on the most promising relaxation, namely what we call ( α, β )-modules.Our initial idea was to allow some “errors” by saying that at most k edges(for some fixed integer k ) could be missing in the complete bipartite subgraphbetween M and N ( M ), denoted ( M, N ( M )), and, symmetrically, that at most k extra edges can exist between M and N ( M ). But by doing so, we loose mostof the nice algebraic properties of modules which yield an underlying partitivefamily. Furthermore, most modular decomposition algorithms are based on thesealgebraic properties [9].A second natural idea is to relax the condition on the complete bipartitesubgraph ( M, N ( M )), for example by asking for a graph that does not containany 2 K (two disjoint edges). Unfortunately, as shown in [37], to test whether agiven graph admits such a decomposition is N P -complete. In fact, in the samework, the authors studied a generalized join decomposition solving a questionraised in [26] about perfection.For all the above reasons and obstacles, we focus on ( α, β )-modules whichmaintain some algebraic properties and thus allow to obtain nice algorithms.Intuitively, we want the reader to think of an ( α, β )-module as a subset ofvertices that almost looks the same from the outside. So, if M is an ( α, β )-module, then for all x, y ∈ M , N ( x ) \ M and N ( y ) \ M are the same, with theexception of at most α + β “errors”, where an error is either a missing edge oran extra edge. We use the integers α and β to bound the number of errors inthe adjacency, according to their type.Formally, we define an ( α, β )-module as follows. Definition 3. An ( α, β ) -module of a graph G = ( V, E ) is a set of vertices M ⊆ V that satisfies ∀ x ∈ V \ M, | M ∩ N ( x ) | ≥ | M | − α or | M ∩ N ( x ) | ≤ β. In other words, M can be turned into a (standard) module by adding at most α edges or deleting at most β edges at each vertex outside M . This notion of missing or extra edges, that we call ( α, β ) -errors , finds appli-cation naturally in various fields, from data compression and exact encodings toapproximation algorithms.Indeed, modular decomposition is often presented as an efficient way to en-code a graph. This encoding property is preserved under the ( α, β )-modules.We want to be able to contract a non-trivial ( α, β )-module (to be preciselydefined later, see Definition 7) into a single vertex while keeping almost theentirety of the original graph, and then apply induction on the decomposition.To this end, for a graph G = ( V, E ), let M be a non-trivial ( α, β )-module with X = N ( M ) and Y = N ( M ). If we want an exact encoding of G , we can contract M into a unique vertex m adjacent to every vertex in X , and non-adjacent toany vertex in Y . We then keep track of the subgraph G ( M ) and the errors thatpotentially arose from the missing edges in ( M, X ) and the extra edges in (
M, Y ).This new encoding has at least | X | · ( | M | − α −
1) edges less than the originalencoding in the worst case and | X | · ( | M | −
1) when M is a module.A second natural and useful application of ( α, β )-modules concerns approx-imation algorithms. Similarly to how cographs are the totally decomposablegraphs with respect to standard modular decomposition, we can define ( α, β )-cographs as the totally decomposable graphs with no ( α, β )-prime graphs (seeDefinition 8). Now consider the classical colouring and independent set programson cographs. The linear time algorithms for these problems both use modulardecomposition. Roughly speaking, the algorithms compute a modular decompo-sition tree, and keep track of the series and parallel internal nodes of the cotreeby scanning the tree from the leaves to the root. Now for α + β ≤
1, we can geta simple 2-approximation algorithm for ( α, β )-cographs for both colouring andindependent set, just by summing over all the ( α, β )- errors. α, β )-Modules
In order to maintain some of the algebraic properties of modules, and avoidrunning into the
N P -complete scenarios previously mentioned, the ( α, β ) gener-alization of modules seems to be a good compromise.We emphasize a few points concerning ( α, β )-modules. Note first that we tol-erate α or β “error-edges” per vertex outside the module, depending on how thisvertex is connected to the ( α, β )-module, and not α + β ) error-edges per module .Secondly, observe that when α = β = 0, we recover the standard definition ofmodules (see Definition 1), which can be rephrased as follows. Definition 4.
A module of a graph G = ( V, E ) is a set of vertices M ⊆ V thatsatisfies ∀ x ∈ V \ M, M ∩ N ( x ) = ∅ or M ∩ N ( x ) = M. Of course we only consider cases for which max( α, β ) < | V | −
1. Fig. 2illustrates an example of a graph with a (1 , a • b • c • d • e • f • g • Fig. 2.
The set { d, e, f } is not a standard module, nor a (1 ,
0) or a (0 , , Proposition 1. If M is an ( α, β ) -module of G , then the following holds.1. M is an ( α ′ , β ′ ) -module of G , for every α ≤ α ′ and β ≤ β ′ .2. M is a ( β, α ) -module of G .3. M is an ( α, β ) -module of every induced subgraph G ( N ) of G with M ⊆ N .4. Every ( α, β ) -module of G ( M ) is an ( α, β ) -module of G .Proof.
1. Taking α ′ and β ′ such that α ≤ α ′ and β ≤ β ′ can only relax the moduleconditions.2. Moving to the complement just interchanges the roles of α and β in thedefinition.3. If the ( α, β )-module conditions are satisfied for all vertices in V ( G ) \ M , thenthey are satisfied for all vertices in V ( G ) \ N for M ⊆ N . Therefore M is an( α, β )-module of the induced subgraph G ( N ).4. Let N be an ( α, β )-module of the subgraph G ( M ). So every vertex in M \ N satisfies the ( α, β )-module conditions. Since M is supposed to be an ( α, β )-module, every vertex in V ( G ) \ M satisfies the ( α, β )-module conditions for M and therefore also for N ⊆ M . Definition 5.
Let G = ( V, E ) be a graph and A ⊆ V be a set of vertices. The α -neighbourhood and β -non-neighbourhood of A are, respectively, N α ( A ) = { x / ∈ A : | N ( x ) ∩ A | ≥ | A | − α } , and N β ( A ) = { x / ∈ A : | N ( x ) ∩ A | ≤ β } . Moreover, if x ∈ N α ( A ) (resp. x ∈ N β ( A ) ), we say that x is an α -neighbour of A (resp. a β -non-neighbour of A ) and that x is α -adjacent (resp. β -non-adjacent to every vertex of A . Definition 6.
Let G = ( V, E ) be a graph and A ⊆ V be a set of vertices. Avertex z / ∈ A is an ( α, β ) -splitter for A if β < | N ( z ) ∩ A | < | A | − α. We denote by S α,β ( A ) the set of ( α, β ) -splitters of A . Hence, a set A is an ( α, β )-module if and only if S α,β ( A ) = ∅ . As an immediateconsequence we have the following easy facts. Lemma 1.
For every graph G = ( V, E ) and every set of vertices A ⊆ V , thefollowing holds.1. N α ( A ) ∪ N β ( A ) ∪ S α,β ( A ) = V \ A .2. If | A | ≥ α + β + 1 , then N α ( A ) ∩ N β ( A ) = ∅ .3. If | A | ≤ α + β + 1 , then S α,β ( A ) = ∅ .4. If | A | = α + β + 1 , then N α ( A ) and N β ( A ) partition V \ A .5. If A is an ( α, β ) -module of G and | A | ≥ α + β + 1 , then N α ( A ) and N β ( A ) partition V \ A .Proof.
1. This directly follows from the definitions of these sets.2. If x ∈ N α ( A ), then | N ( x ) ∩ A | ≥ | A | − α ≥ β + 1 and thus x / ∈ N β ( A ).3. If x ∈ S α,β ( A ), then | N ( z ) ∩ A | < | A | − α < β + 1, a contradiction.4. We have N α ( A ) ∩ N β ( A ) = ∅ by Item 2, and S α,β ( A ) = ∅ by Item 3. Theresult then follows from Item 1.5. This follows from Items 1 and 2. ⊓⊔ Lemma 2.
For every graph G = ( V, E ) and every set of vertices A ⊆ V , if | A | ≤ α + β + 1 , then A is an ( α, β ) -module of G .Proof. Using Lemma 1.3, A admits no ( α, β )-splitter and is thus an ( α, β )-module of G . ⊓⊔ It thus seems that the subsets of size α + β + 1 are crucial to the study ofthis new decomposition. In fact, if A is such a set, then for every vertex z / ∈ A ,we have either z ∈ N α ( A ) or z ∈ N β ( A ), but not both (Lemma 1.3). Lemma 3.
If a vertex s is an ( α, β ) -splitter for a set A , then s is also an ( α, β ) -splitter for every set B ⊇ A with s / ∈ B .Proof. Let s be an ( α, β )-splitter of A . We thus have β < | N ( s ) ∩ A | < | A | − α .Now, if A ⊆ B and s / ∈ B , then we have β < | N ( s ) ∩ A | ≤ | N ( s ) ∩ B | . Similarly,since N ( s ) \ B ⊇ N ( s ) \ A , we have | N ( s ) ∩ B | < | B | − α . Therefore, we get β < | N ( s ) ∩ B | < | B | − α and s is an ( α, β )-splitter for B . ⊓⊔ Theorem 1.
For every graph G = ( V, E ) , the family of ( α, β ) -modules of G satisfies the following.1. The set V is an ( α, β ) -module of G , and every set A ⊆ V with | A | ≤ α + β +1 is an ( α, β ) -module of G .2. If A and B are two ( α, β ) -modules of G , then A ∩ B is an ( α, β ) -module of G . Moreover, the ( α, β ) -splitters of A \ B and B \ A can only belong to A ∩ B . Proof.
1. This directly follows from Definition 3 and Lemma 2.2. First notice that if both A and B are ( α, β )-modules considered in Item 1,then so do A ∩ B , A \ B and B \ A . Suppose then that the cardinality ofboth A and B is at least α + β + 2 and at most | V | − A ∩ B has an ( α, β )-splitter outside of A ∪ B then, by Lemma 3, A and B would also have an ( α, β )-splitter, a contradiction. If A ∩ B has an ( α, β )-splitter in B \ A (resp. in A \ B ) then, by Lemma 3, again A (resp. B ) wouldhave an ( α, β )-splitter. Therefore, A ∩ B is an ( α, β )-module of G .Let us now consider A \ B . If A \ B has an ( α, β )-splitter in B \ A then, byLemma 3, A would have a ( α, β )-splitter as well. The same conclusion arisesfor splitters outside of A ∪ B . Hence, the only possible ( α, β )-splitters for A \ B and, similarly, for B \ A , are in A ∩ B . ⊓⊔ Since the family of ( α, β )-modules is closed under intersection, it yields anotion of graph convexity . Given a set A , we can compute the minimal (underinclusion) ( α, β )-module M ( A ) that contains A , with strictly more than α + β +1elements, thus computing a modular closure via ( α, β )-splitters. Furthermore,the dual cases of (1 , , Definition 7. An ( α, β ) -module M of a graph G = ( V, E ) is a trivial ( α, β ) -module if either M = V or | M | ≤ α + β + 1 . Definition 8.
A graph is an ( α, β ) -prime graph if it has only trivial ( α, β ) -modules. Observe here that when α = β = 0, trivial ( α, β )-modules are exactly trivial(standard) modules, and ( α, β )-prime graphs are exactly prime graphs.From Lemma 2, we directly get the following result. Corollary 1.
A graph G = ( V, E ) with | V | ≤ α + β + 1 has only trivial ( α, β ) -modules. However, we want to distinguish “truly” ( α, β )-prime graphs and “degenerate”( α, β )-prime graphs.
Definition 9.
A graph G = ( V, E ) is ( α, β ) -degenerate if | V | ≤ α + β + 2 . Let us say that a non-trivial ( α, β )-module A is a minimal non-trivial( α, β )-module if every ( α, β )-module strictly contained in A is trivial. Thefollowing result directly follows from this definition. Proposition 2. If A and B are overlapping minimal non-trivial ( α, β ) -modulesof a graph G , then A ∩ B is a trivial ( α, β ) -module of G . From Theorem 1, we get the following result.
Corollary 2.
For every graph G , the family of ( α, β ) -modules of G is a Moorefamily. In the standard setting, if X and Y are modules, then X ∪ Y , X \ Y , and Y \ X are also modules [9,23]. Unfortunately, this does not always hold in the( α, β ) setting. But we can prove a weaker result, namely that we still have an“almost partitive” family. Theorem 2.
Let A and B be two non-trivial overlapping ( α, β ) -modules of agraph G . If | A ∩ B | ≥ α + β + 1 , then A ∪ B and A∆B are (2 α, β ) -modulesof G .Proof. Let z ∈ V \ B . We have S α,β ( B ) = ∅ since B is an ( α, β )-module,and | B | ≥ α + β + 1 since B is non-trivial. Therefore, by Lemma 1.5, N α ( B )and N β ( B ) partition V \ B . Suppose z ∈ N α ( B ). Then, z has at most α non-neighbours in A ∩ B and thus at least β + 1 neighbours in A ∩ B . This gives z ∈ N α ( A ) since A is an ( α, β )-module.Consider first A ∪ B . In the worst case, z has at most α non-neighbours in A \ B and at most α non-neighbours in B \ A . Using the same reasoning on N β ( B ), we get that A ∪ B is a (2 α, β )-module.Consider now A∆B . We just have to further consider the case of vertices in A ∩ B . Also in the worst case, errors arise when a given vertex z ∈ A ∩ B has α (resp. β ) errors in A \ B and α (resp. β ) errors in B \ A . Therefore A∆B is a(2 α, β )-module. ⊓⊔ α, β )-Modular Decomposition TreesDefinition 10. Let G = ( V, E ) be a graph. Two disjoint sets of vertices A, B ⊆ V , with | A | , | B | ≥ α + β + 1 , are said to be α -connected if A ⊆ N α ( B ) and B ⊆ N α ( A ) . Similarly, they are said to be β -non-connected if A ⊆ N β ( B ) and B ⊆ N β ( A ) . In other words, A and B are α -connected if every vertex in A is an α -neighbour of B and every vertex in B is an α -neighbour of A . They are β -non-connected if every vertex in A is a β -non-neighbour of B and every vertexin B is a β -non-neighbour of A . Definition 11.
Let G = ( V, E ) be a graph. – An ( α, β ) -modular partition of G is a partition of V into ( α, β ) -modules. – For | V | ≥ α + β + 3 , we say that G admits an α -series (resp. a β -parallel ) decomposition if there exists an ( α, β ) -modular partition of V , P = { V , . . . , V k } , such that1. ∃ j ∈ [ k ] such that | V j | ≥ α + β + 1 , and2. ∀ i, j ∈ [ k ] , i = j , V i and V j are α -connected (resp. β -non-connected). – For | V | ≥ α + β +3 , we say that G admits an ( α, β ) -prime decomposition if there exists an ( α, β ) -modular partition of V , P = { V , . . . , V k } , with k ≥ such that ∀ i ∈ [ k ] , V i is maximal (under inclusion),2. ∃ j ∈ [ k ] such that | V j | ≥ α + β + 1 , and3. there exist two pairs ( i, j ) and ( p, q ) , i = j , p = q , such that V i and V j are α -connected while V p and V q are β -non-connected. Using Proposition 1.2 we have the following obvious property.
Property 1.
A graph G admits an α -series decomposition if and only if G admitsan α -parallel decomposition. Proposition 3. If A and B are two disjoint ( α, β ) -modules of a graph G with | A | , | B | ≥ α + β + 1 , then N α ( A ) ⊇ B and N β ( B ) ⊇ A are mutually exclusive.Proof. Suppose to the contrary that N α ( A ) ⊇ B and N β ( B ) ⊇ A .Let m A,B be the number of edges in G joining A and B . We thus have | B | · ( | A | − α ) ≤ m A,B ≤ | A | · β = ⇒ | A | · | B | < β · | A | + α · | B | . Now, let | A | = α + β + a and | B | = α + β + b , with a, b ∈ N ∗ . We then get α · a + β · b + ab ≤ , a contradiction. ⊓⊔ Furthermore it could be the case that A is α -connected to B and that B isneither α -connected to A nor β -non-connected to A . Corollary 3. If A and B are two disjoint ( α, β ) -modules of a graph G with | A | ≥ α + β + 1 and α ≥ , then the inclusions N α ( A ) ⊇ B and N β ( B ) ⊇ A aremutually exclusive.If A and B are two disjoint ( α, β ) -modules of a graph G with | B | ≥ α + β + 1 and β ≥ , then the inclusions N α ( A ) ⊇ B and N β ( B ) ⊇ A are mutuallyexclusive.Proof. If | A | ≥ α + β + 1 and | B | ≥ α + β + 1, this directly follows fromProposition 3. Otherwise, the proof is similar to the proof of Proposition 3, bysimply considering the two extreme cases, i.e., b ≤ α ≥ a ≥
1, or a ≤ β ≥ b ≥ Proposition 4.
The three cases of Definition 11 are mutually exclusive for agiven partition into ( α, β ) -modules.Proof. Clearly the α -series and β -parallel decompositions are each exclusive withthe ( α, β )-prime case.Now let us prove that α -series and β -parallel cases are exclusive. If α = β = 0 it is just Gallai’s theorem. So we suppose that there exists an ( α, β )-modular partition P = { V , . . . , V k } which is both an α -series and a β -paralleldecomposition. Suppose, without loss of generality, that | V | ≥ α + β + 1 (weknow that such a set exists). Now we have two cases to consider.If α ≥ V is α -connected to V then V cannotbe β -non-connected to V .Similarly, if β ≥
1, again using Corollary 3, if V is β -non-connected to V then V cannot be α -connected to V . ⊓⊔ Open Problem 1
Is this result also true for α -series and β -paralleldecompositions based on different partitions? (The interesting case iswhen all intersections between the two partitions have size bounded by α + β .) It should be noticed that in the case of an α -series (resp. a β -parallel) de-composition, a union of V i ’s is not necessarily an ( α, , β )-module). Such a property is always true only for standard modules. Definition 12.
Using the terminology of [12] for combinatorial decompositions,we will say that a graph G = ( V, E ) is ( α, β ) -brittle if every subset of V is an ( α, β ) -module. Of course, ( α, α edges), and (0 , β )-independent graphs (i.e., independent sets with at most β edges) are ( α, β )-brittle, but they are not the only obvious ones; any path P k isalso (1 , G with | V | ≤ α + β +2 are ( α, β )-brittle,and we called them ( α, β )-degenerate to distinguish them from the “truly” ( α, β )-prime graphs. All these remarks raise the question of the characterization of( α, β )-brittle graphs. Clearly, any graph G with minimum degree at least | V | − α or maximum degree at most β is ( α, β )-brittle. Open Problem 2
Can we characterize the ( α, β ) -brittle graphs? Using Definition 11 and Proposition 1.4, and mimicking the case of standardmodular decomposition, we may define an ( α, β )-modular decomposition tree asfollows.
Definition 13. An ( α, β ) -modular decomposition tree of a graph G =( V, E ) is a tree whose nodes are labelled with ( α, β ) -modules ordered by inclu-sion with four types of nodes:1. α -series,2. β -parallel,3. ( α, β ) -prime, and4. ( α, β ) -degenerate.Each level of the tree corresponds to a partition of V , starting with { V } atthe root, and such that the leaves correspond to a partition of V into ( α, β ) -degenerate nodes. Such an ( α, β )-modular decomposition tree completely describes what we callan ( α, β )-modular decomposition of a graph G . In the standard modular decomposition setting, the notion of strong modules,i.e., modules that do not overlap any other module, is quite central. In the ( α, β )-modular decomposition setting, observe that there are no strong ( α, β )-modulesother than { V } and the singletons { v : v ∈ V } . This comes from the fact thatwhen max { α, β } ≥
1, every subset of vertices of size 2 is a trivial ( α, β )-module.Now, assume there is a standard strong module A = V with | A | >
1. By takingany vertex v ∈ A and any vertex u ∈ V \ A , we get an ( α, β )-module of size 2which overlaps A .Recall that the Modular Decomposition Theorem [19] (Gallai’s Theorem)says that every graph admits a unique modular decomposition tree. In the ( α, β )setting, we have the following weaker form of Gallai’s Theorem. Theorem 3.
Every graph G = ( V, E ) with | V | ≥ α + β + 3 admits an ( α, β ) -modular decomposition tree, for every ( α, β ) with ≤ α, β ≤ | V | − .Proof. Let G = ( V, E ) be an arbitrary graph. If G is an ( α, β )-prime graph then G admits a trivial ( α, β )-modular decomposition tree T ( G ), with a root labelled( α, β )-prime. The leaves of T ( G ) are the children of the root, where every leaf isassociated with a partition of V into sets of α + β + 1 vertices.Suppose now that G is not an ( α, β )-prime graph. Then G admits at leastone non-trivial ( α, β )-module. Let M be any maximal (under inclusion) non-trivial ( α, β )-module, and let R be the set of remaining vertices: R = V \ M .We consider two cases, depending on the size of R .1. | R | ≤ α + β .If R = N α ( M ), then we build a tree with a root r labelled ( α, β )-series, andtwo children, T ( G ( M R (labelled ( α, β )-degenerate).If R = N β ( M ), then we build a tree with a root r labelled ( α, β )-paralleland two children, T ( G ( M R (labelled ( α, β )-degenerate).Finally, in the remaining case, both N α ( M ) and N β ( M ) are non-empty.We then build a tree with a root r labelled ( α, β )-prime and three children, T ( G ( M N α ( M ) and N β ( M ), the last two nodes being both labelled( α, β )-degenerate.We then apply induction on G ( M ) to obtain T ( G ( M | R | ≥ α + β + 1.We simply apply the same reasoning on G ( V \ M ).In both cases, we eventually obtain a partition P = { M , . . . , M k } of V with k ≥
3. The partition P is an ( α, β )-modular partition in which only M k couldhave less than α + β vertices.We then check whether P induces an α -series decomposition or a β -paralleldecomposition. If not, we say that P induces an ( α, β )-prime decomposition.We then build a tree with a root labelled ( α, β )-prime (resp. α -series, β -parallel) and whose children are the ( α, β )-modular decomposition trees of thesubgraphs G ( M i ), 1 ≤ i ≤ k , and apply induction on the G ( M i )’s, using Propo-sition 1.4. ⊓⊔ Maximal ( α, β )-modules may overlap, which unfortunately means, as it isalso illustrated in the examples of the next sections, that this ( α, β )-modulardecomposition tree is not unique. Although such a decomposition tree can pro-vide an exact encoding of the graph (if the ( α, β )-errors are traced), it doesnot provide an encoding of all existing ( α, β )-modules, see Figure 3 for instance.Furthermore, although the above proof is constructive, Theorem 3 does not leadto an efficient algorithm for computing an ( α, β )-modular decomposition tree.In fact, we do not know any polynomial algorithm to compute M , a maximalnon-trivial ( α, β )-module, to start with. In the next section we will see that itcould be the case that no such algorithm exists. α -Series and β -Parallel Operations Let us focus on the α -series and β -parallel decompositions. Definition 14. An ( α, β ) -cograph is a graph that is totally decomposable withrespect to α -series and β -parallel decompositions until we reach ( α, β ) -degeneratesubgraphs only. Using Definition 14 above, it is clear that standard cographs are preciselythe (0 , ( α, β )-cotree the tree corresponding to an( α, β )-modular decomposition of an ( α, β )-cograph. Although several different( α, β )-cotrees may be associated with an ( α, β )-cograph, the following proposi-tion shows that we can always find such an ( α, β )-cotree having only α -seriesand β -parallel internal nodes. Proposition 5.
A graph is an ( α, β ) -cograph if and only if it admits an ( α, β ) -cotree having only α -series and β -parallel internal nodes.Proof. We construct the ( α, β )-cotree of an ( α, β )-cograph G recursively as fol-lows. Suppose G has an α -series and β -parallel decomposition with partition P = { V , . . . V k } . Notice first that these two operations are exclusive since everypart V i has at least α + β + 1 vertices, and thus two parts V i and V j cannotbe both α -connected and β -non-connected. The ( α, β )-cotree is then obtainedby taking a root r , labelled α -series, with k children T ( G ( V i )), 1 ≤ i ≤ k , andconstruct each subtree T ( G ( V i )) by applying induction on the subgraph G ( V i ),using Proposition 1.4. ⊓⊔ Consider the two examples illustrated in Figures 3 and 4. Fig. 3 shows a (1 , H that admits a unique (1 , , G that admits two different (1 , G in Fig. 4 by an isomorphic copy of G , and repeat this process, we can build a(1 , exponentially many different (1 , α = 0 and β = 1, the problem of recognizing if agraph admits a (0 , abc de fh g (1,0)-series (0,1)-par.(0,1)-par.(0,1)-deg. { a, b } (0,1)-deg. { c, d } (0,1)-deg. { e, f } (0,1)-deg. { g, h } Fig. 3.
A (1 , , H on the left. Notice that H is not acograph since it contains two induced P ’s: { a, b, c, d } and { e, f, g, h } . a b c defx y (0,1)-parallel (0,1)-para. (0,1)-para.(1,1)-deg. { a,b } (1,1)-deg. { c,d } (1,1)-deg. { e,f } (1,1)-deg. { x,y } (0,1)-parallel (0,1)-para. (0,1)-para.(1,1)-deg. { a,b } (1,1)-deg. { f,x } (1,1)-deg. { c,y } (1,1)-deg. { e,d } Fig. 4.
The graph G on the top is a (1 , , V induced by the leavesof the (1 , equivalent to finding a matching cut set in a graph, i.e., an edge cut which is amatching, a well-known problem studied in [4,10,33].Unfortunately however, it turns out –as one might expect– that finding sucha matching cut set in an arbitrary graph is an NP-complete problem, as shownby Chv´atal in [10]. Theorem 4 ([10]).
Deciding if a graph has a matching cut set is NP-complete.
Definition 15.
Let G = ( V, E ) be a graph and P = { V , . . . , V k } be the partitionof V associated with an ( α, β ) -modular decomposition of G . If every union ofparts from { V , . . . , V k } is an ( α, β ) -module, we say that such a decomposition isan ( α, β ) -modular brittle decomposition of G . From Theorem 4, we get the following theorem.
Theorem 5.
For a graph G = ( V, E ) with | V | ≥ , deciding if G admits a (0 , -parallel (resp. a (1 , -series) brittle decomposition is NP-complete.Proof. Suppose G admits a (0 , P = { V , . . . , V k } . Since it is a brittle decomposition, S
Definition 16. An ( α, β ) -modular decomposition tree of a graph G is minimal if the first partition level has a minimal number of parts, among all possible ( α, β ) -modular decomposition trees of G . We can consider two decision problems, depending on whether α and β arepart of the input or not. Minimal modular decomposition
Input:
Two positive integers α, β and a graph G . Question:
Does G admit a minimal ( α, β )-modular decomposition? Minimal ( α, β ) -modular decomposition Input:
A graph G . Question:
Does G admit a minimal ( α, β )-modular decomposition?Let us prove the NP-hardness of Minimal modular decomposition . Proposition 6.
For every pair of integers ( α, β ) = (0 , , and every graph G with | V ( G ) | > α + β ) , G cannot admit an ( α, β ) -series decomposition anda ( α, β ) -parallel decomposition both in two parts, say { A , A } and { B , B } ,respectively.Proof. First we notice that any two parts from different partitions must overlap.Indeed, suppose for example that A ⊆ B . We then necessarily have B ⊆ A .We then get that a vertex x ∈ A must be α -connected and β -non-connected to B , a contradiction.Since V ( G ) = ∪ i,j ∈{ , } A i ∩ B j and | V ( G ) | > α + β ), one of these four sets,say C = A ∩ B , has at least α + β + 1 elements. But then, every x ∈ A ∩ B is α -connected and β -non-connected to C , a contradiction. ⊓⊔ Therefore if G admits a minimal ( α, β )-modular decomposition with onlytwo parts, then it is either an ( α, β )-series decomposition or an ( α, β )-paralleldecomposition, but not both. Corollary 4.
Minimal modular decomposition is NP-complete.Proof.
Since computing a minimal (0 , G has matching cut set or not. But by Theorem 4 such a compu-tation is NP-complete. So for α = 0 and β = 1 the problem is NP-complete. ⊓⊔ We also think that Minimal ( α, β )-modular decomposition is NP-completefor every ( α, β ) = (0 , α, β )-modular decomposition tree is NP-hard, which webelieve to be true, since, as shown in Figure 4, a given graph may have twodifferent (0 , Conjecture 1
For every ( α, β ) with max { α, β } > , finding an ( α, β ) -modular decomposition of a graph G is an NP-complete problem. In the above-mentioned work [10], Chv´atal showed that the matching cut setproblem is NP-complete on graphs with maximum degree four, and polynomialon graphs with maximum degree three. In fact the problem of finding a perfectmatching cut set is also NP-hard [25].On the other hand, computing a matching cut set in the following graphclasses is polynomial: – graphs with maximum degree three [10], – weakly chordal graphs and line-graphs [33], – Series-Parallel graphs [36], – claw-free graphs and graphs with bounded clique-width, as well as graphswith bounded treewidth [4], – graphs with diameter 2 [5], and – ( K , , K , + e )-free graphs [28].Therefore, to check whether a graph in any of the above classes is a (1 , , t , taking two graphsobtained in t − Matching Composition Networks in [42], contains all hypercubes, aswell as all crossed, twisted and M¨obius hypercubes. In general,
P M G ( k ) is afamily of graphs recursively defined, that starts with all connected graphs on k vertices and, at every step, add any graph that can be obtained by selecting twographs within the family having the same order and joining them with a perfectmatching.More formally, the family PMG (4), for instance, is defined as follows.
Definition 17 (PMG (4) ). We start with the following seven connected graphson four vertices: – P , C , K , K , , – a triangle with a pending edge, – two triangles having an edge in common.At every step, a new graph is obtained from two graphs in the family having thesame order by joining them with a perfect matching. Knowing that
PMG (1) =
PMG (2) and that they contain hypercubes,crossed, twisted and M¨obius hypercubes, we end this section with the follow-ing recognition problem.
Open Problem 3
Given a graph G = ( V, E ) with | V | = 2 n , what isthe complexity of recognizing whether G belongs to PMG ( k ) or to oneof its non-trivial subclasses? α, β )-Prime GraphsProposition 7. The only (1 , -prime graph of order is C .Proof. Let C = [ a, b, c, d, e ]. To prove that C is a (1 , , A , the remaining vertex not in A is connectedby exactly two edges to A . A systematic study of all the other graphs of order 5(including the bull) shows that they all have a non-trivial (1 , ⊓⊔ Algorithm 1:
Computing minimal ( α, β )-modules.
Input:
A graph G = ( V, E ) and a set A ⊆ V with | A | ≥ α + β + 2. Output: M ( A ), the unique minimal ( α, β )-module that contains A M ← A , i ← S ← { x ∈ V \ M : β < | N ( x ) ∩ M | < | M |− α } ; while S = ∅ do i ← i + 1; M i ← M i − ∪ S ; S ← { x ∈ V \ M i : β < | N ( x ) ∩ M i | < | M i |− α } ; M ( A ) ← M i ; Notice here that the Petersen graph can be obtained by a (0 , C .Obviously, we have the following inclusion: For all α ≤ α ′ and all β ≤ β ′ , thefamily of ( α, β )-prime graphs is included in the family of ( α ′ , β ′ )-prime graphs.But can we improve this result? In the standard setting, the prime graphs arenested. In particular, P is the smallest prime graph, and all primes on n verticescontain a prime subgraph on either n − n − Open Problem 4
Are the ( α, β ) -prime graphs nested? α, β )-modules Despite the negative hardness results in the previous sections, we shall now ex-amine how to compute all minimal ( α, β )-modules of a given graph in polynomialtime. As mentioned earlier, non-trivial ( α, β )-modules have strictly more than α + β + 2 elements; and since they are closed under intersection, ( α, β )-moduleshave an underlying graph convexity, and thus (see Algorithm 1), we can computethe minimal ( α, β )-module M ( A ) that contains a given set A with | A | > α + β +2,by computing a modular closure via ( α, β )-splitters. In fact, we build a seriesof subsets M i that starts with M = A and satisfies M i ⊆ M i +1 for every i ≥ Proposition 8.
Algorithm 1 computes the unique minimal ( α, β ) -module thatcontains A in O ( m · n ) time.Proof. If A is an ( α, β )-module, then in line 2, S = ∅ ; otherwise, all the elementsof S have to be added into M ( A ). In other words, using Lemma 3, there is no( α, β )-module M such that : A ( M ( A ∪ S . At the end of the while loop,either M i = V or we have found a non-trivial ( α, β )-module that contains A .This algorithm obviously runs in O ( m · n ) time. ⊓⊔ Algorithm 2:
Computing minimal ( α, β )-modules.
Input:
A graph G and A ⊆ V ( G ) with | A | ≥ α + β + 2. Output: M ( A ), the minimal ( α, β )-module that contains A OP EN ← A ; M ( A ) ← ∅ ; foreach u ∈ V do CLOSED ( u ) ← F ALSE ; edge ( u ) ← non - edge ( u ) ← while OP EN = ∅ do Select a vertex z from OP EN and delete z from OP EN ; Add z to M ( A ); CLOSED ( z ) ← T RUE ; foreach u neighbour of z do if CLOSED ( u ) = F ALSE and u / ∈ M ( A ) then edge ( u ) ← edge ( u ) + 1; if β < edge ( u ) and α < non - edge ( u ) then Add u to OPEN foreach v non-neighbour of z do if CLOSED ( u ) = F ALSE and u / ∈ M ( A ) then non - edge ( v ) ← non - edge ( v ) + 1; if β < edge ( u ) and α < non - edge ( u ) then Add v to OPEN Algorithm 2 proposes a different implementation that uses a graph searchapproach to compute the minimal ( α, β )-module containing A . This will allowus to achieve a linear running time. Theorem 6.
Algorithm 2 can be implemented in O ( m + n ) time.Proof. We can implement Algorithm 2 as a kind of a graph search, using analgorithm less naive than Algorithm 1. Algorithm 2 also computes the minimal( α, β )-module that contains A in a graph search manner.At the end of Algorithm 2, the set M ( A ) contains a minimal ( α, β )-modulethat contains A . At first glance, this algorithm requires O ( n ) operations, sincefor each vertex we must consider all its neighbours and all its non-neighbours.However, if we use a partition refinement technique as defined in [24], startingwith a partition of the vertices as P = { A, V \ A } . We then keep in the samepart, B ( i, j ), vertices x, y with edge ( x ) = edge ( y ) = i and non - edge ( x ) = non - edge ( y ) = j . This way, when visiting a vertex z , it suffices to compute B ′ ( i + 1 , j ) = B ( i, j ) ∩ N ( z ), and B ′′ ( i, j + 1) = B ( i, j ) − N ( z ) , for each part B ( i, j ) in the current partition. This can be done in O ( | N ( z ) | )time.It should be noted that the parts need not to be sorted in the current parti-tion, and we may have different parts with the same (edge, non-edge) values.Algorithm 2 can thus be implemented in O ( m + n ) time. ⊓⊔ Theorem 7.
Using Algorithm 2, one can compute all the minimal non-trivial ( α, β ) -modules of a given graph in O ( m · n α + β +1 ) time.Proof. To do so, it suffices to use Algorithm 2 starting from every subset of α + β + 2 vertices. There exists O ( n α + β +2 ) such subsets. This will therefore giveus a straight O ( n α + β +2 ) running time. However, we can use partition refinementto out advantage by using the neighbourhood of one vertex exactly once. Buta vertex can belong to at most n α + β +1 parts in the partition refinement, whichyields an algorithm with O ( m · n α + β +1 ) running time. ⊓⊔ Remark : If we consider the (0 ,
0) case, i.e., the standard case, in Algorithm 2,we find the implementation of the algorithm in [27], which also computes all theminimal modules in O ( m · n ) time – to be compared to the original one in O ( n )time. Corollary 5.
Using theorem 7, one can compute a covering of V with an over-lapping family of minimal ( α, β ) -modules in O ( m · n α + β +2 ) time. Moreover, theoverlapping of any two members of the obtained covering is bounded by α + β + 1 .Proof. Using theorem 7, we can compute an overlapping family of minimal ( α, β )-modules in O ( m · n α + β +1 ) time. But this family can possibly not be a full coveringof V since some vertices may not belong to any minimal non-trivial ( α, β )-module.To obtain a full covering, we then simply add the remaining vertices as singletons. ⊓⊔ Corollary 5 can be very interesting if we are looking for overlapping commu-nities in social networks, where the overlapping is bounded by α + β + 1.Going a step further, we can use Theorem 2 and merge every pair A, B of( α, β )-modules with | A ∩ B | ≥ α + β + 1, either by keeping A ∪ B as a (2 α, β )-module, or by computing M ( A ∪ B ), the minimal ( α, β )-module that contains A ∪ B . This depends however on the structure of the maximal ( α, β )-modules,and unfortunately we do not know yet under which conditions there exists aunique partition into maximal ( α, β )-modules. Corollary 6.
Checking if a graph is ( α, β ) -prime can be done in O ( m · n α + β +1 ) time.Proof. Easy using Theorem 7. ⊓⊔ In this section, let G = ( X, Y, E ) be a bipartite graph with parts X and Y . Byallowing α + β errors in the decomposition, ( α, β )-modules can be made up withvertices from both X and Y . However, in some applications, we are forced toconsider X and Y separately. Consider for instance the setting where X and Y represent the sets of customers and products, or the sets of DNA sequences andorganisms, in which case one would want to find regularities on each side of thebipartition. Definition 18.
For a given bipartite graph G = ( X, Y, E ) , we let F α,β ( X ) = { M : M is an ( α, β ) -module of G and M ⊆ X } . Note that X is not always an ( α, β ) -module of G . Proposition 9.
For every two sets
A, B ∈ F α,β ( X ) , A ∩ B , A \ B and B \ A are all in F α,β ( X ) .Proof. Using Theorem 1, the only ( α, β )-splitters of the sets A \ B and B \ A must belong to A ∩ B ; but since A, B ⊆ X , and X is an independent set, this isnot possible. ⊓⊔ It should be noticed here that for A ⊆ X , the minimal ( α, β )-module thatcontains A does not always belong to F α,β ( X ), since we may have to add ( α, β )-splitters from Y . Therefore we have to use an algorithmic approach differentfrom those developed in the previous section in order to compute F α,β ( X ). Definition 19.
Two sets
A, B ⊆ V , A = B , with | A | = | B | = α + β + 1 , aresaid to be false ( α, β ) -twin (resp. true ( α, β ) -twin ) in G if they satisfy thefollowing three conditions:1. A ∪ B is an ( α, β ) -module,2. ∀ x ∈ A , x ∈ N α ( B ) (resp. x ∈ N β ( B ) ),3. ∀ y ∈ B , y ∈ N α ( A ) (resp. y ∈ N β ( A ) ). Observe that A and B are false ( α, β )-twin sets in G if and only if A and B are true ( α, β )-twin sets in G . Proposition 10.
Being (true or false) ( α, β ) -twin is an equivalence relation onsubsets of vertices. When applying Definition 19 to bipartite graphs, we obviously only have false( α, β )-twin sets.
Proposition 11.
A set M ⊆ X is an ( α, β ) -module if and only if M is a unionof false ( α, β ) -twin sets.Proof. Let
A, B ⊆ M , with | A | = | B | = α + β + 1. Pick any vertex z ∈ Y . If z ∈ N α ( A ), then z ∈ N α ( M ) and therefore z ∈ N α ( B ). Therefore, A and B arefalse ( α, β )-twin sets since they are both included in X .The converse directly follows from Definition 19. ⊓⊔ Consequently, in terms of ( α + β + 1)-tuples, the sets of false ( α, β )-twin setspartition the ( α + β +1)-tuples. Furthermore, using the notion of false ( α, β )-twinsets, we obtain the following theorem (recall that for a graph G , F α,β is the setof its ( α, β )-modules whose elements are in X ). Theorem 8.
For a given bipartite graph G = ( X, Y, E ) , the maximal elementsof F α,β ( X ) can be computed in O ( n α + β ( n + m )) time. Proof.
To do so, we first build an auxiliary bipartite graph, G ′ = ( A , Y, E ( G ′ )),which represents the labelled incidence graph of the ( α + β + 1)-tuples of verticesof X . The set of vertices of G ′ is thus the set A of these ( α + β + 1)-tuples,By Lemma 1.4, we know that every such tuple T yields a partition of Y into N α ( T ) and N β ( T ). The set of edges of G ′ is then defined by setting, for every T ∈ A and y ∈ Y , T y ∈ E ( G ′ ) if and only if y ∈ N α ( T ) , which implies T y / ∈ E ( G ′ ) if and only if y ∈ N β ( T ) . Since every vertex in Y belongs to at most O ( n α + β ) tuples from A , the numberof edges in E ( G ′ ) is in O ( m · n α + β ).Given the auxiliary graph G ′ , we now partition A into false twins. To this aim,we use every vertex in Y to refine A with respect to ( α, β )-neighbourhood. Thiscan be done in O ( n α + β +1 + n α + β · m ) time, using standard partition refinementtechniques [24].Let Q = {A , . . . A k } be such a partition. We prove the following claim. Claim:
No element of F α,β can contain two ( α + β + 1) -tuples from differentparts of Q .Proof. Let A i ∈ A i , A j ∈ A j , with i = j , and S be a subset of X such that A i ∪ A j ⊆ S . Since i = j , there is a vertex y ∈ Y such that (w.l.o.g.) A i ∈ N α ( y )and A j ∈ N β ( y ). Hence we have | A i | − α ≤ | S ∩ N ( y ) | ≤ | S | − | A j | + β, which gives β + 1 ≤ | S ∩ N ( y ) | ≤ | S | − α − , and thus y is an ( α, β )-splitter for S . ⊓⊔ Therefore, to find the maximal elements of F α,β , we can restrict the searchto the A i ’s. Let us now examine how to generate them. To this aim, we definea labelling λ that assigns to each ordered pair ( y, A ), with y ∈ Y and A ∈ A , asubset of A as follows. – If yA ∈ E ( G ′ ) and a , . . . , a k , k ≤ α , are the vertices from A non adjacentto y , then we set λ ( y, A ) = { a , . . . , a k } . – Symmetrically, if yA / ∈ E ( G ′ ) and a , . . . , a h , h ≤ β , are the vertices from A adjacent to y , then we set λ ( y, A ) = { a , . . . , a h } .This labelling can be done while constructing the graph G ′ .Then, a maximal element F of F α,β is just a maximal union of elements ofsome A i , 1 ≤ i ≤ k , satisfying the following: – For every vertex y ∈ Y , • if every element of A i is adjacent to y , then | ∪ A ∈ F λ ( y, A ) | ≤ α , • otherwise, | ∪ A ∈ F λ ( y, A ) | ≤ β .Note that all vertices in A i are false twins, since the graph G ′ is bipartite,and therefore connected the same way to Y .To produce these maximal sets, we start with α = β = 0, in which casethe only maximal module has an empty label. Let M denote this module and M , = { M } denote the set of maximal elements at this step. We then increaseeither α or β by one, and recursively compute the new set, M α +1 ,β or M α,β +1 ,of maximal elements from the previously computed set M α,β (note that everymaximal ( α, β )-module is contained in a maximal ( α + 1 , β )-module and in amaximal ( α, β + 1)-module as well).For α = β = 0, M is unique. For α = β = 1, there are at most | Y | maximal(1 , F , . Hence, there are at most | Y | α + β maximal ( α, β )-modulesin F α,β . This computation is therefore bounded in the whole by( α + β )( Σ i = α + βi =0 | Y | i ) · ( | X | α + β +1 ) , which is in the order of O (( | Y | α + β +1 ) · ( | X | α + β +1 )). ⊓⊔ Note that these maximal elements of F α,β ( X ) may overlap. It remains to testthe quality of the covering obtained on some real data graphs. We leave this assomething to explore for data analysts. Before we conclude, we want first to expose the reader to a different way toapproach the approximation of modules. k -splitter Modules: An Alternate Approximation Another natural way to approach the problem of approximating modules is byrestricting the number of splitters a module can have. Recall that in the standardmodular decomposition setting, a splitter of a module M in a graph G = ( V, E )is a vertex v ∈ V \ M such that there exists at least two vertices a, b ∈ M with av ∈ E and bv / ∈ E . By restricting the number of splitters outside a module, weget the following definition – which intuitively just allows at most k “errors” interms of connectivity. Definition 20.
For a given graph G = ( V, E ) , a subset M of V is a k -splittermodule if M has at most k splitters. Notice then that by setting k = 0 in the above definition, we recover thestandard modular decomposition setting [23], i.e., for every x ∈ V \ M , either M ∩ N ( x ) = ∅ or M ∩ N ( x ) = M . So, for this approximate setting, we willnecessarily only consider the case k < | V ( G ) | − Proposition 12. If M is a k -splitter for G , then the following holds.1. M is a k ′ -splitter module for G , for every k ′ ≥ k .2. M is a k -splitter module for G .3. If s is a splitter for M , then s is also a splitter for every set M ′ ⊇ M with s / ∈ M ′ . Proposition 13.
The family of k -splitter modules of a graph G = ( V, E ) satis-fies the following.1. Every set A ⊆ V with | A | ≤ or | A | ≥ | V | − k is a k -splitter module of G .(We call such a set A a trivial k -splitter module .)2. For every two k -splitter modules A, B ⊆ V of G with A ∩ B = ∅ , A ∪ B is a k -splitter module of G .3. For every two k -splitter modules A, B ⊆ V of G with A ∩ B = ∅ , A ∩ B is a k -splitter module of G .Proof.
1. This follows from the definition.2. There cannot be a splitter of A ∪ B that is not a splitter of either A or B since A ∩ B = ∅ . We get the 2 k in the worst case, when both A and B havetwo disjoint sets of k splitters outside of A ∪ B .3. A splitter of A ∩ B in V \ ( A ∪ B ) is a splitter of both A and B . A splitterof A ∩ B in A is a splitter of B and a splitter of A ∩ B in B is a splitterof A . Therefore, the number of splitters of A ∩ B is at most the sum of thenumbers of splitters of A and B , i.e., 2 k . ⊓⊔ Proposition 14. If A and B are two non-trivial k -splitter modules of a graph G = ( V, E ) , then A \ B is a ( k + | A ∩ B | ) -module of G .Proof. There are at most k splitters of A \ B in V \ A , and at most | A ∩ B | splitters of A \ B in A ∩ B . ⊓⊔ Proposition 15.
For a graph G = ( V, E ) and a subset A ⊆ V , there may existdifferent minimal (under inclusion) k -splitter modules containing A .Proof. Suppose A admits k + 1 splitters. Every one of these k + 1 splitters canbe added to A in order to obtain a k -splitter module. ⊓⊔ In conclusion, this approximation variation is not closed under intersection,unfortunately. There was no way to define some sort of convexity, and thus noeasy way to define a closure operator with this notion, which is why we havefocused our study on ( α, β )-modules instead. In this work, we introduce a new notion of modular decomposition relaxation.This notion of ( α, β )-module yields many interesting questions, both from atheoretical or practical point of view. Standard modular decomposition is toorestrictive for graphs that arise from real data; do ( α, β )-modules indeed oftenarise in this setting? We believe this relaxation of modular decomposition can def-initely find applications in practice. On the theory side, this new combinatorialdecomposition may help to better understand graph structuration that can beobtained when grouping vertices that have similar neighbourhood. Such an ideahas been successfully used with the notion of twin-width [3,2]. Furthermore it isrelated to fundamental combinatorial objects as for example matching cutsetsand their generalization. Our work leaves many interesting questions open (fiveopen questions and one conjecture), as the study of ( α, β )-prime graphs for in-stance. We have also exhibited new classes of graphs, such as the (1 , α, β )-modules some hierarchy of modules. However,perhaps a better way to decompose a graph is to first compute the families ofminimal modules with small values of α and β , and then consider a hierarchy ofoverlapping families. References
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