The RankWidth of EdgeColored Graphs
Abstract
Cliquewidth is a complexity measure of directed as well as undirected graphs. Rankwidth is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertexminor relation. We discuss an extension of the notion of rankwidth to edgecolored graphs. A colored graph is a graph where the arcs are colored with colors from the set . There is not a natural notion of rankwidth for colored graphs. We define two notions of rankwidth for them, both based on a coding of colored graphs by edgecolored graphs where each edge has exactly one color from a field and named respectively rankwidth and birankwidth. The two notions are equivalent to cliquewidth. We then present a notion of vertexminor for colored graphs and prove that colored graphs of bounded rankwidth are characterised by a finite list of colored graphs to exclude as vertexminors. A cubictime algorithm to decide whether a colored graph has rankwidth (resp. birankwidth) at most , for fixed , is also given. Graph operations to check MSOLdefinable properties on colored graphs of bounded rankwidth are presented. A specialisation of all these notions to (directed) graphs without edge colors is presented, which shows that our results generalise the ones in undirected graphs.
keywords:
rankwidth; cliquewidth; local complementation; vertexminor; excluded configuration; 2structure; sigmasymmetry.and
1 Introduction
Cliquewidth [5, 12] is a complexity measure for edgecolored graphs, i.e., graphs where edges are colored with colors from a finite set. CliqueWidth is more general than treewidth [33] because every class of graphs of bounded treewidth has bounded cliquewidth and the converse is false (complete undirected graphs have cliquewidth and unbounded treewidth) [12]. Cliquewidth is an interesting complexity measure in algorithmic design. In fact every property expressible in monadic secondorder logic (MSOL for short) can be checked in lineartime, provided the cliquewidth expression is given, on every graph that has small cliquewidth [11]. This result is important in complexity theory because many NPcomplete problems are MSdefinable properties, e.g., colorability. However, it is NPcomplete to check if a graph has cliquewidth at most when is part of the input [17]. It is still open whether this problem is polynomial for fixed .
In their investigations of a recognition algorithm for undirected graphs of cliquewidth at most , for fixed , Oum and Seymour [31] introduced the notion of rankdecomposition and associated complexity measure rankwidth, of undirected graphs. Rankwidth is defined in a combinatorial way and is equivalent to the cliquewidth of undirected graphs in the sense that a class of graphs has bounded cliquewidth if and only if it has bounded rankwidth [31]. But, being defined in a combinatorial way provides to rankwidth better algorithmic properties than cliquewidth, in particular:

for fixed , there exists a cubictime algorithm that decides whether the rankwidth of an undirected graph is at most and if so, constructs a rankdecomposition of width at most [22];
Since cliquewidth and rankwidth of undirected graphs are equivalent, one way to check MSOL properties in undirected graphs of small rankwidth is to transform a rankdecomposition into a cliquewidth expression [31]. However, an alternative characterization of rankwidth in terms of graph operations has been proposed in [9]. It is thus possible to solve MSOL properties in graphs of small rankwidth by using directly the rankdecomposition. This later result is important in a practical point of view because it avoids the exponent, that cannot be avoided [4, 32], when transforming a rankdecomposition into a cliquewidth expression.
Another advantage of rankwidth over cliquewidth is that it is invariant with respect to the vertexminor relation (no such notion, except for induced subgraph relation, is known for cliquewidth), i.e., if is a vertexminor of , then the rankwidth of is at most the rankwidth of [29]. Moreover, every class of undirected graphs of bounded rankwidth is characterised by a finite list of undirected graphs to exclude as vertexminors [29]. This later result generalises the one of Robertson and Seymour on undirected graphs of bounded treewidth [33].
Despite all these positive results of rankwidth, the fact that cliquewidth is defined for graphs  directed or not, with edge colors or not  is an undeniable advantage over rankwidth. It is thus natural to ask for a notion of rankwidth for edgecolored graphs or at least for directed graphs without edge colors. Courcelle and Oum suggested in [13] a definition of rankwidth for directed graphs as follows: Courcelle [6] described a graph transformation from (directed) graphs to undirected bipartite graphs so that , for some functions and ; the rankwidth of a (directed) graph is defined as the rankwidth of . This definition can be extended to edgecolored graphs by using a similar coding (see [7, Chapter 6]). This definition gives a cubictime algorithm that approximates the cliquewidth of edgecolored graphs. Another consequence is the proof of a weak version of the Seese’s conjecture for edgecolored graphs [13]. However, this definition suffers from the following drawback: a vertexminor of does not always correspond to a coding of an edgecolored graph and similarly for the notion of pivotminor (see for instance [20, 29] for the definition of pivotminor of undirected graphs).
We investigate in this paper a better notion of rankwidth for edgecolored graphs. However, there is no unique natural way to extend rankwidth to edgecolored graphs. We are looking for a notion that extends the one on undirected graphs and that can be used for directed graphs without edge colors. For that purposes, we will define the notion of sigmasymmetric matrices, which generalizes the notion of symmetric and skewsymmetric matrices. We then use this notion to represent edgecolored graphs by matrices over finite fields and derive, from this representation, a notion of rankwidth, called rankwidth, that generalises the one of undirected graphs. We also define another notion of rankwidth, called birankwidth. We prove that the two parameters are equivalent to cliquewidth.
We then define a notion of vertexminor for edgecolored graphs that extends the one on undirected graphs. We prove that rankwidth and birankwidth are invariant with respect to this vertexminor relation. We give a characterisation of edgecolored graphs of bounded rankwidth by excluded configurations. This result generalises the one on undirected graphs [29]. A generalisation of the pivotminor relation is also presented.
The cubictime recognition algorithm by Hlinný and Oum [22] can be adapted to give for fixed , a cubictime algorithm that decides if a given edgecolored graph has rankwidth (resp. birankwidth) at most and if so, outputs an optimal rankdecomposition.
The two notions of rankwidth of edgecolored graphs are specialised to directed graphs without colors on edges. All the results specialised to them.
The paper is organized as follows. In Section 2 we give some preliminary definitions and results. We recall in particular the definition of rankwidth of undirected graphs. The first notion of rankwidth of edgecolored graphs, called rankwidth, is studied in Section 3. We will define the notion of vertexminor and pivotminor, and prove that edgecolored graphs of bounded rankwidth are characterised by a finite list of edgecolored graphs to exclude as vertexminors (resp. pivotminors). A cubictime recognition algorithm and a specialisation to directed graphs are also presented. We define our second notion of rankwidth for edgecolored graphs called birankwidth in Section 4. We also specialise it to directed graphs. In Section 5 we introduce some algebraic graph operations that generalise the ones in [9]. These operations will be used to characterise exactly the two notions of rankwidth. They can be seen as alternatives to cliquewidth operations for solving MSOL properties. We conclude by some remarks and open questions in Section 6.
This paper is related to a companion paper where the authors introduce a decomposition of edgecolored graphs on a fixed field [24]. This decomposition plays a role similar to the split decomposition for the rankwidth of undirected graphs. Particularly we show that the rank width of an edgecolored graph is exactly the maximum over the rankwidth over all edgecolored prime graphs in the decomposition, and we give different characterisations of egdecolored graphs of rankwidth one.
2 Preliminaries
For two sets and , we let be the set . The powerset of a set is denoted by . We often write to denote the set . The set of natural integers is denoted by .
We denote by and the binary operations of any field and by and the neutral elements of and respectively. For every prime number and every positive integer , we denote by the finite field of characteristic and of order . We recall that they are the only finite fields. We refer to [27] for our field terminology.
For sets and , an matrix is a matrix where the rows are indexed by elements in and columns indexed by elements in . For an matrix , if and , we let be the submatrix of where the rows and the columns are indexed by and respectively. We let be the matrix rankfunction (the field will be clear from the context). We denote by the transpose of a matrix . The order of an matrix is defined as . We often write matrix to denote a matrix of order . For positive integers and , we let be the null matrix and the identity matrix, or respectively and when the size is clear in the context.
We use the standard graph terminology, see for instance [15]. A graph is a couple where is the set of vertices and is the set of edges. A graph is said to be oriented if implies , and it is said undirected if implies . An edge between and in an undirected graph is denoted by (equivalently ). For a graph , we denote by , called the subgraph of induced by , the graph ; we let be the subgraph . The degree of a vertex in an undirected graph is the cardinal of the set . Two graphs and are isomorphic if there exists a bijection such that if and only if . We call an isomorphism between and . All graphs are finite and loopfree (i.e. for every , ).
A tree is an acyclic connected undirected graph. In order to avoid confusions in some lemmas, we will call nodes the vertices of trees. The nodes of degree are called leaves and the set of leaves in a tree is denoted by . A subcubic tree is an undirected tree such that the degree of each node is at most . A tree is rooted if it has a distinguished node , called the root of . For convenience, we will consider a rooted tree as an oriented graph such that the underlying graph is a tree, and such that all nodes are reachable from the root by a directed path. For a tree and an edge of , we let denote the graph .
Let be a (possibly infinite) set that we call the colors. A graph is a tuple where is a graph and is a function. Its associated underlying graph is the graph . Two graph and are isomorphic if there is a isomorphism between and such that for every , . We call an isomorphism between and . We let be the set of graphs for a fixed color set . Even though we authorise infinite color sets in the definition, most of results in this article are valid only when the color set is finite. It is worth noticing that an edgeuncolored graph can be seen as an edgecolored graph where all the edges have the same color.
Remark 1 (Multiple colors per edge)
In our definition, an edge in a graph can only have one color. However, this is not restrictive because if in an edgecolored graph an edge can have several colors from a set , we just extend to .
Remark 2 (structures and edgecolored graphs)
A structure [16] is a pair where is a finite set and is an equivalence relation on the set . Every structure can be seen as a colored graph where is an equivalence class of and for every edge . Equivalently, every graph can be seen as a structure where if and only if and all the nonedges in are equivalent with respect to .
A parameter on is a function that is invariant under isomorphism. Two parameters on , say and , are equivalent if there exist two integer functions and such that for every edgecolored graph , .
The cliquewidth, denoted by , is a graph parameter defined by Courcelle et al. [5, 12]. Most of the investigations concern edgeuncolored graphs. However, its edgecolored version has been investigated these last years (see [3, 18]). Note that the cliquewidth is also defined in more general case where edges can have several colors.
We finish these preliminaries by the notion of terms. Let be a set of binary and unary function symbols and a set of constants. We denote by the set of finite wellformed terms built with . Notice that the syntactic tree of a term is rooted.
A context is a term in having a single occurrence of the variable (a nullary symbol). We denote by the set of contexts. We denote by the particular context . If is a context and a term, we let be the term in obtained by substituting for in .
2.1 RankWidth and VertexMinor of Undirected Graphs
Despite the interesting algorithmic results [11], cliquewidth suffers from the lack of a recognition algorithm. In their investigations for a recognition algorithm, Oum and Seymour introduced the notion of rankwidth [31], which approximates the cliquewidth of undirected graphs. Let us first define some notions.
Let be a finite set and a function. We say that is symmetric if for any ; is submodular if for any , .
A layout of a finite set is a pair of a subcubic tree and a bijective function . For each edge of , the connected components of induce a bipartition of , and thus a bipartition of (we will omit the sub or supscript when the context is clear).
Let be a symmetric function and a layout of . The width of each edge of is defined as and the width of is the maximum width over all edges of . The width of is the minimum width over all layouts of .
Definition 3 (Rankwidth of undirected graphs [29, 31])
For every undirected graph , we let be its adjacency matrix where if and only if . For every graph , we let where , where is the matrix rank over . This function is symmetric. The rankwidth of an undirected graph , denoted , is the width of .
RankWidth has several structural and algorithmic results, see for instance [9, 22, 29]. In particular, for fixed , there exists a cubictime algorithm for recognizing undirected graphs of rankwidth at most [22]. Moreover, rankwidth is related to a relation on undirected graphs, called vertexminor.
Definition 4 (Local complementation, Vertexminor [29])
For an undirected graph and a vertex of , the local complementation at , denoted by , consists in replacing the subgraph induced on the neighbors of by its complement. A graph is a vertexminor of a graph if can be obtained from by applying a sequence of local complementations and deletions of vertices.
Authors of [2, 20, 29] also introduced the pivot operation on an edge , denoted by . An interesting theorem relating rankwidth and the notion of vertexminor is the following.
Theorem 5 ([29])
For every positive integer , there exists a finite list of undirected graphs such that an undirected graph has rankwidth at most if and only if it does not contain as vertexminor any graph isomorphic to a graph in .
In the next section, we define the notion of rankwidth of edgecolored graphs and generalize Theorem 5 to them.
3 RankWidth of Symmetic Graphs
We want a notion of rankwidth for edgecolored graphs that generalises the one on undirected graphs. For that purposes, we will identify each color by an nonzero element of a field. This representation will allow us to define the rankwidth of edgecolored graphs by using rank matrices.
Let be a field, and let (where is the zero of ). One can note that there is a natural bijection between the class of graphs and the class of graphs with complete underlying graph (replace every nonedge by an edge of color ). From now on, we do not distinguish these two classes, and we let for all .
We can represent every graph by a matrix such that for every with , and for every .
Let be a bijection. We recall that is an involution if for all . We call a sesquimorphism if is an involution, and the mapping is an automorphism. It is worth noticing that if is a sesquimorphism, then and for every , (i.e. is an automorphism for the addition). Moreover, we have the following notable equalities.
Proposition 6
If is a sesquimorphism, then
A graph is symmetric if the underlying graph is undirected, and for every arc , if and only if . Clearly, if is a symmetric graph, then . We denote by (respectively ) the set of graphs (respectively symmetric graphs). Note that .
To represent a graph, one can take an injection from to for a large enough field . Notice that the representation is not unique: on one hand, several incomparable fields are possible for , and on the other hand, the representation depends on the injection from to . For example, oriented graphs can be represented by a graph or by a graph (see Section 3.4). Two different representations can give two different rankwidth parameters, but the two parameters are equivalent when is finite (direct consequence of Proposition 16).
Let be a finite field of characteristic and order . We will prove that every graph can be seen as a symmetric graph for some sesquimorphism , where is an algebraic extension of of order . Let us first make some observations.
Lemma 7
There exists an element in such that the polynomial has no root in .
Proof. There exist distinct polynomials of the form , . We first notice that or cannot be a root of , for any . Now, two such polynomials cannot have a common root. Assume the contrary and let be a root of and of with . Then , i.e. since , a contradiction. Since and cannot be the roots of any of the polynomials, we have at most possible roots. Therefore, there exists a such that has no root in .∎
We can now construct an algebraic extension of the finite field . Let such that has no root in and let be isomorphic to the field (i.e. is the finite field of characteristic and order ). Let . Then every element of is a polynomial on of the form where . Moreover, is a root of in .
We let and be in . Notice that and .
Lemma 8
We have the following equalities:
To every pair of elements in , we associate an element in by letting where, for every , .
Lemma 9
is a bijection.
For the sesquimorphism in , we let where . One easily verifies that for all .
Lemma 10
is an automorphism.
Proof. An easy computation shows that . For the product, we have:
and  
By Lemma 8, , and . This concludes the proof of the lemma. ∎
For every graph , we let be the graph where, for every two distinct vertices and ,
Proposition 11
The mapping from to is a bijection and for every graph , is symmetric. Moreover, for two graphs and , and are isomorphic if and only if and are isomorphic.
Nevertheless, two different mappings can give two different rankwidth parameters. But again, since is finite, the parameters are equivalent.
If is infinite, a mapping from to is not always possible with the previous construction. For example, a mapping is possible from to with and (where ), but the construction fails for since the complexes are algebraically closed.
From now on, we will focus our attention to sigmasymmetric graphs. In Section 3.1 we define the notion of rankwidth. The notion of vertexminor for graphs is presented in Section 3.2 and we prove that sigmasymmetric graphs of rankwidth at most are characterised by a finite list of sigmasymmetric graphs to exclude as vertexminors. We prove in Section 3.3 that graphs of rankwidth at most , for fixed , can be recognised in cubictime when is finite. A specialisation to graphs without colors on edges is presented in Section 3.4.
3.1 RankWidth of symmetric Graphs
Along this section, we let be a fixed field (of characteristic and of order if is finite), and we let be a fixed sesquimorphism. We recall that if is a graph, we denote by the matrix where:
Definition 12 (CutRank Functions)
The cutrank function of a symmetric graph is the function where for all .
Lemma 13
For every symmetric graph , the function is symmetric and submodular.
We first recall the submodular inequality of the matrix rankfunction.
Proposition 14
[29, Proposition 4.1] Let be an matrix over a field . Then for all and ,
Proof of Lemma 13. Let and be subsets of . We let and . We first prove the first statement.
We let be the matrix where . Since is a sesquimorphism, the mapping is an automorphism and then . But, . Then, .
For the second statement, we have by definition and Proposition 14,
Since and , the second statement holds. ∎
Definition 15 (rankwidth)
The rankwidth of a symmetric graph , denoted by , is the width of .
This definition generalises the one for undirected graphs. If we let be the identity automorphism on , every undirected graph is a symmetric graph. Moreover, for every undirected graph , the functions and are equal. It is then clear that the definition of rankwidth given in Section 2.1 coincides with the one of rankwidth.
One can easily verify that the rankwidth of a symmetric graph is the maximum of the rankwidth of its maximum connected components. The following proposition, which says that rankwidth and cliquewidth are equivalent when is finite, has an easy proof. We omit it because its proof is an easy adaptation of the one comparing rankwidth and cliquewidth of undirected graphs [31, Proposition 6.3].
Proposition 16
Let be a symmetric graph. Then, .
3.2 VertexMinor and PivotMinor
Bouchet generalised in [2] the notion of local complementation to all graphs (undirected or not). We recall that a graph is a graph and then is represented by a matrix over where if and only if . A local complementation at of is the graph represented by the matrix over where . This definition coincides with the one on undirected graphs when is undirected. We will extend it to graphs. We say that in is compatible if .
Definition 17 (local complementation)
Let in . Let be a graph and a vertex of . The local complementation at of is the graph represented by the matrix where:
One can easily verify that for every graph and every vertex of , the adjacency matrix of is obtained by modifying the submatrix induced by the neighbors of . Then for every vertex of .
Definition 18 (locally equivalent, vertexminor)
A graph is locally equivalent to a graph if is obtained by applying a sequence of local complementations to with . We call a vertexminor of if for some and is locally equivalent to . Moreover, is a proper vertexminor of if .
In this section, we are interested in symmetric graphs, thus we have to restrict ourselves to a subset of local complementations which preserve the symmetry. We now prove that local complementation is well defined on symmetric graphs when is compatible.
Lemma 19
Let be a symmetric graph and let be compatible. Then every local complementation of is also symmetric.
Proof. Let for some compatible . It is sufficient to prove that for any .
Definition 20 (locallyequivalent, vertexminor)
A graph is locallyequivalent to a symmetric graph if is obtained by applying a sequence of localcomplementations to with compatibles . We call a vertexminor of if for some and is locallyequivalent to . Moreover, is a proper vertexminor of if .
Note that if no compatible exists, is a vertexminor of if and only if is an induced subgraph of .
Remark 21
The following lemma proves that localcomplementation does not increase rankwidth.
Lemma 22
Let be a symmetric graph and a vertex of . For every subset of ,
Proof. We can assume that since is a symmetric function (Lemma 13). For each , the localcomplementation at results in adding a multiple of the row indexed by to the row indexed by . Precisely, we obtain by adding to . This operation is repeated for all . In each case, the rank of the matrix does not change. Hence, .∎
Unfortunately, such a compatible does not always exist. For instance, if the field is and is such that (see Section 3.4), no compatible does exist. We present now an other graph transformation which is defined for every couple .
Definition 23 (Pivotcomplementation)
Let be a symmetric graph, and and two vertices of such that . The pivotcomplementation at of is the graph represented by the matrix where for every , and for every with :
A graph is pivotequivalent to a graph if is obtained by applying a sequence of pivotcomplementations to . We call a pivotminor of if for some and pivotequivalent to . Moreover, is a proper pivotminor of if .
Note that if . In the case of undirected graphs (), this definition coincides with the pivotcomplementation of undirected graphs [29]. The following lemma shows that this transformation is well defined.
Lemma 24
Let be a symmetric graph and let be an edge of . Then is also symmetric.
Proof. Let , with . If , then
If , then:
Finally:
Similarly to Lemma 22, the following lemma proves that pivot complementation does not increase rankwidth.
Lemma 25
Let be a symmetric graph and an edge of . For every subset of :
Proof. Let