The Rank-Width of Edge-Colored Graphs
Clique-width is a complexity measure of directed as well as undirected graphs. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss an extension of the notion of rank-width to edge-colored graphs. A -colored graph is a graph where the arcs are colored with colors from the set . There is not a natural notion of rank-width for -colored graphs. We define two notions of rank-width for them, both based on a coding of -colored graphs by edge-colored graphs where each edge has exactly one color from a field and named respectively -rank-width and -bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for -colored graphs and prove that -colored graphs of bounded -rank-width are characterised by a finite list of -colored graphs to exclude as vertex-minors. A cubic-time algorithm to decide whether a -colored graph has -rank-width (resp. -bi-rank-width) at most , for fixed , is also given. Graph operations to check MSOL-definable properties on -colored graphs of bounded rank-width 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:rank-width; clique-width; local complementation; vertex-minor; excluded configuration; 2-structure; sigma-symmetry.
Clique-width [5, 12] is a complexity measure for edge-colored graphs, i.e., graphs where edges are colored with colors from a finite set. Clique-Width is more general than tree-width  because every class of graphs of bounded tree-width has bounded clique-width and the converse is false (complete undirected graphs have clique-width and unbounded tree-width) . Clique-width is an interesting complexity measure in algorithmic design. In fact every property expressible in monadic second-order logic (MSOL for short) can be checked in linear-time, provided the clique-width expression is given, on every graph that has small clique-width . This result is important in complexity theory because many NP-complete problems are MS-definable properties, e.g., -colorability. However, it is NP-complete to check if a graph has clique-width at most when is part of the input . It is still open whether this problem is polynomial for fixed .
In their investigations of a recognition algorithm for undirected graphs of clique-width at most , for fixed , Oum and Seymour  introduced the notion of rank-decomposition and associated complexity measure rank-width, of undirected graphs. Rank-width is defined in a combinatorial way and is equivalent to the clique-width of undirected graphs in the sense that a class of graphs has bounded clique-width if and only if it has bounded rank-width . But, being defined in a combinatorial way provides to rank-width better algorithmic properties than clique-width, in particular:
for fixed , there exists a cubic-time algorithm that decides whether the rank-width of an undirected graph is at most and if so, constructs a rank-decomposition of width at most ;
Since clique-width and rank-width of undirected graphs are equivalent, one way to check MSOL properties in undirected graphs of small rank-width is to transform a rank-decomposition into a clique-width expression . However, an alternative characterization of rank-width in terms of graph operations has been proposed in . It is thus possible to solve MSOL properties in graphs of small rank-width by using directly the rank-decomposition. 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 rank-decomposition into a clique-width expression.
Another advantage of rank-width over clique-width is that it is invariant with respect to the vertex-minor relation (no such notion, except for induced sub-graph relation, is known for clique-width), i.e., if is a vertex-minor of , then the rank-width of is at most the rank-width of . Moreover, every class of undirected graphs of bounded rank-width is characterised by a finite list of undirected graphs to exclude as vertex-minors . This later result generalises the one of Robertson and Seymour on undirected graphs of bounded tree-width .
Despite all these positive results of rank-width, the fact that clique-width is defined for graphs - directed or not, with edge colors or not - is an undeniable advantage over rank-width. It is thus natural to ask for a notion of rank-width for edge-colored graphs or at least for directed graphs without edge colors. Courcelle and Oum suggested in  a definition of rank-width for directed graphs as follows: Courcelle  described a graph transformation from (directed) graphs to undirected bipartite graphs so that , for some functions and ; the rank-width of a (directed) graph is defined as the rank-width of . This definition can be extended to edge-colored graphs by using a similar coding (see [7, Chapter 6]). This definition gives a cubic-time algorithm that approximates the clique-width of edge-colored graphs. Another consequence is the proof of a weak version of the Seese’s conjecture for edge-colored graphs . However, this definition suffers from the following drawback: a vertex-minor of does not always correspond to a coding of an edge-colored graph and similarly for the notion of pivot-minor (see for instance [20, 29] for the definition of pivot-minor of undirected graphs).
We investigate in this paper a better notion of rank-width for edge-colored graphs. However, there is no unique natural way to extend rank-width to edge-colored 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 sigma-symmetric matrices, which generalizes the notion of symmetric and skew-symmetric matrices. We then use this notion to represent edge-colored graphs by matrices over finite fields and derive, from this representation, a notion of rank-width, called -rank-width, that generalises the one of undirected graphs. We also define another notion of rank-width, called -bi-rank-width. We prove that the two parameters are equivalent to clique-width.
We then define a notion of vertex-minor for edge-colored graphs that extends the one on undirected graphs. We prove that -rank-width and -bi-rank-width are invariant with respect to this vertex-minor relation. We give a characterisation of edge-colored graphs of bounded -rank-width by excluded configurations. This result generalises the one on undirected graphs . A generalisation of the pivot-minor relation is also presented.
The cubic-time recognition algorithm by Hlinný and Oum  can be adapted to give for fixed , a cubic-time algorithm that decides if a given edge-colored graph has -rank-width (resp. -bi-rank-width) at most and if so, outputs an optimal rank-decomposition.
The two notions of rank-width of edge-colored 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 rank-width of undirected graphs. The first notion of rank-width of edge-colored graphs, called -rank-width, is studied in Section 3. We will define the notion of vertex-minor and pivot-minor, and prove that edge-colored graphs of bounded -rank-width are characterised by a finite list of edge-colored graphs to exclude as vertex-minors (resp. pivot-minors). A cubic-time recognition algorithm and a specialisation to directed graphs are also presented. We define our second notion of rank-width for edge-colored graphs called -bi-rank-width in Section 4. We also specialise it to directed graphs. In Section 5 we introduce some algebraic graph operations that generalise the ones in . These operations will be used to characterise exactly the two notions of rank-width. They can be seen as alternatives to clique-width 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 edge-colored graphs on a fixed field . This decomposition plays a role similar to the split decomposition for the rank-width of undirected graphs. Particularly we show that the rank width of an edge-colored graph is exactly the maximum over the rank-width over all edge-colored prime graphs in the decomposition, and we give different characterisations of egde-colored graphs of rank-width one.
For two sets and , we let be the set . The power-set 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  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 sub-matrix of where the rows and the columns are indexed by and respectively. We let be the matrix rank-function (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 . 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 sub-graph of induced by , the graph ; we let be the sub-graph . 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 loop-free (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 sub-cubic 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 edge-uncolored graph can be seen as an edge-colored 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 edge-colored graph an edge can have several colors from a set , we just extend to .
Remark 2 (-structures and edge-colored graphs)
A -structure  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 non-edges 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 edge-colored graph , .
The clique-width, denoted by , is a graph parameter defined by Courcelle et al. [5, 12]. Most of the investigations concern edge-uncolored graphs. However, its edge-colored version has been investigated these last years (see [3, 18]). Note that the clique-width 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 well-formed 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 Rank-Width and Vertex-Minor of Undirected Graphs
Despite the interesting algorithmic results , clique-width suffers from the lack of a recognition algorithm. In their investigations for a recognition algorithm, Oum and Seymour introduced the notion of rank-width , which approximates the clique-width 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 sub-cubic 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 sup-script 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 .
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 rank-width of an undirected graph , denoted , is the -width of .
Rank-Width has several structural and algorithmic results, see for instance [9, 22, 29]. In particular, for fixed , there exists a cubic-time algorithm for recognizing undirected graphs of rank-width at most . Moreover, rank-width is related to a relation on undirected graphs, called vertex-minor.
Definition 4 (Local complementation, Vertex-minor )
For an undirected graph and a vertex of , the local complementation at , denoted by , consists in replacing the sub-graph induced on the neighbors of by its complement. A graph is a vertex-minor of a graph if can be obtained from by applying a sequence of local complementations and deletions of vertices.
Theorem 5 ()
For every positive integer , there exists a finite list of undirected graphs such that an undirected graph has rank-width at most if and only if it does not contain as vertex-minor any graph isomorphic to a graph in .
In the next section, we define the notion of rank-width of edge-colored graphs and generalize Theorem 5 to them.
3 -Rank-Width of -Symmetic -Graphs
We want a notion of rank-width for edge-colored graphs that generalises the one on undirected graphs. For that purposes, we will identify each color by an non-zero element of a field. This representation will allow us to define the rank-width of edge-colored 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 non-edge 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 sesqui-morphism if is an involution, and the mapping is an automorphism. It is worth noticing that if is a sesqui-morphism, then and for every , (i.e. is an automorphism for the addition). Moreover, we have the following notable equalities.
If is a sesqui-morphism, 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 rank-width 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 sesqui-morphism , where is an algebraic extension of of order . Let us first make some observations.
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 .
We have the following equalities:
To every pair of elements in , we associate an element in by letting where, for every , .
is a bijection.
For the sesqui-morphism in , we let where . One easily verifies that for all .
is an automorphism.
Proof. An easy computation shows that . For the product, we have:
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 ,
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 rank-width 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 sigma-symmetric -graphs. In Section 3.1 we define the notion of -rank-width. The notion of vertex-minor for -graphs is presented in Section 3.2 and we prove that sigma-symmetric -graphs of -rank-width at most are characterised by a finite list of sigma-symmetric -graphs to exclude as vertex-minors. We prove in Section 3.3 that -graphs of -rank-width at most , for fixed , can be recognised in cubic-time when is finite. A specialisation to graphs without colors on edges is presented in Section 3.4.
3.1 Rank-Width 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 sesqui-morphism. We recall that if is a -graph, we denote by the -matrix where:
Definition 12 (Cut-Rank Functions)
The -cut-rank function of a -symmetric -graph is the function where for all .
For every -symmetric -graph , the function is symmetric and submodular.
We first recall the submodular inequality of the matrix rank-function.
[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 sesqui-morphism, 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 (-rank-width)
The -rank-width 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 rank-width given in Section 2.1 coincides with the one of -rank-width.
One can easily verify that the -rank-width of a -symmetric -graph is the maximum of the -rank-width of its maximum connected components. The following proposition, which says that -rank-width and clique-width are equivalent when is finite, has an easy proof. We omit it because its proof is an easy adaptation of the one comparing rank-width and clique-width of undirected graphs [31, Proposition 6.3].
Let be a -symmetric -graph. Then, .
3.2 Vertex-Minor and Pivot-Minor
Bouchet generalised in  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 sub-matrix induced by the neighbors of . Then for every vertex of .
Definition 18 (locally equivalent, vertex-minor)
A -graph is locally equivalent to a -graph if is obtained by applying a sequence of -local complementations to with . We call a vertex-minor of if for some and is locally equivalent to . Moreover, is a proper vertex-minor 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.
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 (-locally-equivalent, -vertex-minor)
A -graph is -locally-equivalent to a -symmetric -graph if is obtained by applying a sequence of -local-complementations to with -compatibles . We call a -vertex-minor of if for some and is -locally-equivalent to . Moreover, is a proper -vertex-minor of if .
Note that if no -compatible exists, is a -vertex-minor of if and only if is an induced subgraph of .
The following lemma proves that -local-complementation does not increase -rank-width.
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 -local-complementation 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 (Pivot-complementation)
Let be a -symmetric -graph, and and two vertices of such that . The pivot-complementation at of is the -graph represented by the -matrix where for every , and for every with :
A -graph is pivot-equivalent to a -graph if is obtained by applying a sequence of pivot-complementations to . We call a pivot-minor of if for some and pivot-equivalent to . Moreover, is a proper pivot-minor of if .
Note that if . In the case of undirected graphs (), this definition coincides with the pivot-complementation of undirected graphs . The following lemma shows that this transformation is well defined.
Let be a -symmetric -graph and let be an edge of . Then is also -symmetric.
Proof. Let , with . If , then
If , then:
Similarly to Lemma 22, the following lemma proves that pivot complementation does not increase -rank-width.
Let be a -symmetric -graph and an edge of . For every subset of :