Comparing Linear Width Parameters for Directed Graphs

12/17/2018
by   Frank Gurski, et al.
0

In this paper we introduce the linear clique-width, linear NLC-width, neighbourhood-width, and linear rank-width for directed graphs. We compare these parameters with each other as well as with the previously defined parameters directed path-width and directed cut-width. It turns out that the parameters directed linear clique-width, directed linear NLC-width, directed neighbourhood-width, and directed linear rank-width are equivalent in that sense, that all of these parameters can be upper bounded by each of the others. For the restriction to digraphs of bounded vertex degree directed path-width and directed cut-width are equivalent. Further for the restriction to semicomplete digraphs of bounded vertex degree all six mentioned width parameters are equivalent. We also show close relations of the measures to their undirected versions of the underlying undirected graphs, which allow us to show the hardness of computing the considered linear width parameters for directed graphs. Further we give first characterizations for directed graphs defined by parameters of small width.

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1 Introduction

A graph parameter is a function that associates with every graph a positive integer. Examples for graph parameters are tree-width [57], clique-width [17], NLC-width [59], and rank-width [55]. Clique-width, NLC-width, and rank-width are equivalent, i.e. a graph has bounded clique-width, if and only if it has bounded NLC-width and that is if and only if it has bounded rank-width. The latter three parameters are more general than tree-width, since graphs of bounded tree-width also have bounded clique-width but even for dense graphs (e.g. cliques) the tree-width is unbounded while the clique-width can be small [36]. Graph classes of bounded width are interesting from an algorithmic point of view since several hard graph problems can be solved in polynomial time by dynamic programming along the tree structure of the input graph, see [2, 4, 39, 45] and [15, 16, 20]. Furthermore such parameters are also interesting from a structural point of view, e.g. in the research of special graph classes [11, 9].

In this paper we consider graph parameters which are defined by the existence of an underlying path-structure for the input graph. These are path-width [56], cut-width [1], linear clique-width [37], linear NLC-width [37], neighbourhood-width [32], and linear rank-width [26]. With the exception of cut-width these parameters can be regarded as restrictions of the above mentioned parameters with underlying tree-structure to an underlying path-structure. The relation between these parameters corresponds to their tree-structural counterparts, since bounded path-width implies bounded linear NLC-width, linear clique-width, neighborhood-width, and linear rank-width. Further the reverse direction is not true in general, see [32]. Such restrictions to underlying path-structures are often helpful to show results for the general parameters, see [21, 22]. These linear parameters are also interesting from a structural point of view, e.g. in the research of special graph classes [26, 31, 41].

Since several problems and applications frequently use directed graphs, during the last years, width parameters for directed graphs have received a lot of attention, see [27, 28] and the two book chapters [6, Chapter 9] and [19, Chapter 6]. Lifting the above mentioned parameters using an underlying tree-structure to directed graphs lead to directed tree-width [43], directed NLC-width [38], directed clique-width [17], and directed rank-width [44].

In this paper we study directed graph parameters which are defined by the existence of an underlying path-structure for the input graph. One of the most famous examples is the directed path-width, which was introduced by Reed, Seymour, and Thomas around 1995 (see [7]) and studied in [7, 58, 49, 48]. Further the cut-width for directed graphs was introduced by Chudnovsky et al. in [12]. Regarding the usefulness of linear width parameters for undirected graphs we introduce the directed linear NLC-width, directed linear clique-width, directed neighborhood-width, and directed linear rank-width. In contrast to the linear width measures for undirected graphs, for directed graphs their relations turn out to be more involved. Table 1 shows some classes of digraphs demonstrating various possible combinations of the listed width measures being bounded and unbounded.

undirected directed DAG CB BS OP TT
cut-width cutw [1] d-cutw [12] 0 0 0
path-width pw [56] d-pw Thomas et al. 0 1 0 0
linear clique-width lcw [37] d-lcw here 2 2 3 2
linear NLC-width lnlcw [37] d-lnlcw here 1 1 3 1
neighbourhood-width nw [32] d-nw here 1 1 2 1
linear rank-width lrw [26] d-lrw here 1 1 2 1
Table 1: Width measures and their values for directed acyclic digraphs (DAG), complete bioriented (CB) digraphs, bioriented stars (BS), oriented paths (OP), and transitive tournaments (TT).

For all these linear width parameters for directed graphs we compare the directed width of a digraph and the undirected width of its underlying undirected graph, which allow us to show the hardness of computing the considered linear width parameters for directed graphs.

In order to classify graph parameters we call two graph parameters

and equivalent, if there are two functions and such that for every digraph the value can be upper bounded by and the value can be upper bounded by . If and are polynomials or linear functions, we call and polynomially equivalent or linearly equivalent, respectively. We show that for general digraphs we have three sets of pairwise equivalent parameters, namely , , and . For digraphs of bounded vertex degree this reduces to two sets and and for semicomplete digraphs of bounded vertex degree all these six graph parameters are pairwise equivalent. With the exception of directed rank-width, the same results are even shown for polynomially and linearly equivalence.

By introducing the class of directed threshold graphs, we give characterizations for graphs defined by parameters of small width.

2 Preliminaries

We use the notations of Bang-Jensen and Gutin [5] for graphs and digraphs.

2.1 Undirected graphs

We work with finite undirected graphs , where is a finite set of vertices and is a finite set of edges. For a vertex we denote by the set of all vertices which are adjacent to in , i.e. . Set is called the set of all neighbors of in or neighborhood of in . The degree of a vertex , denoted by , is the number of neighbors of vertex in , i.e. . The maximum vertex degree is . A graph is a subgraph of graph if and . If every edge of with both end vertices in is in , we say that is an induced subgraph of digraph and we write . For some graph class we define as the set of all graphs such that no induced subgraph of is isomorphic to a member of .

Special Undirected Graphs

We recall some special graphs. By

, we denote a path on vertices and by

, we denote a cycle on vertices. Further by

, we denote a complete graph on vertices and by

a complete bipartite graph on vertices.

2.2 Directed graphs

A directed graph or digraph is a pair , where is a finite set of vertices and

is a finite set of ordered pairs of distinct

111Thus we do not consider directed graphs with loops. vertices called arcs. For a vertex , the sets and are called the set of all out-neighbours and the set of all in-neighbours of . The outdegree of , for short, is the number of out-neighbours of and the indegree of , for short, is the number of in-neighbours of in . The maximum out-degree is and the maximum in-degree is . The maximum vertex degree is . A digraph is a subdigraph of digraph if and . If every arc of with both end vertices in is in , we say that is an induced subdigraph of digraph and we write . For some digraph class we define as the set of all digraphs such that no induced subdigraph of is isomorphic to a member of .

Let be a digraph.

  • is edgeless if for all , , none of the two pairs and belongs to .

  • is a tournament if for all , , exactly one of the two pairs and belongs to .

  • is semicomplete if for all , , at least one of the two pairs and belongs to .

  • is (bidirectional) complete if for all , , both of the two pairs and belong to .

Omitting the directions

For some given digraph , we define its underlying undirected graph by ignoring the directions of the edges, i.e. .

Orientations

There are several ways to define a digraph from an undirected graph . If we replace every edge by

  • one of the arcs and , we denote as an orientation of . Every digraph which can be obtained by an orientation of some undirected graph is called an oriented graph.

  • one or both of the arcs and , we denote as a biorientation of . Every digraph which can be obtained by a biorientation of some undirected graph is called a bioriented graph.

  • both arcs and , we denote as a complete biorientation of . Since in this case is well defined by we also denote it by . Every digraph which can be obtained by a complete biorientation of some undirected graph is called a complete bioriented graph.

Special directed graphs

We recall some special directed graphs. By

we denote a directed path on vertices and by

we denote a directed cycle on vertices. Further let

be a bidirectional complete digraph on vertices. The -power graph of a digraph is a graph with the same vertex set as . There is an arc in if and only if there is a directed path from to of length at most in . An oriented forest (tree) is the orientation of a forest (tree). A digraph is an out-tree (in-tree) if it is an oriented tree in which there is exactly one vertex of indegree (outdegree) zero. A directed acyclic digraph (DAG for short) is a digraph without any , as subdigraph.

3 Linear width parameters for directed graphs

A layout of a graph is a bijective function . For a graph , we denote by the set of all layouts for . Given a layout we define for the vertex sets

The reverse layout , for , is defined by , .

3.1 Directed path-width

The path-width (pw) for undirected graphs was introduced in [56]. The notion of directed path-width was introduced by Reed, Seymour, and Thomas around 1995 (cf. [7]) and relates to directed tree-width introduced by Johnson, Robertson, Seymour, and Thomas in [43].222Please note that there are some works which define the path-width of a digraph in a different and not equivalent way by using the path-width of , see Section 7.

Definition 3.1 (directed path-width)

Let be a digraph. A directed path-decomposition of is a sequence of subsets of , called bags, such that the following three conditions hold true.

  1. ,

  2. for each there is a pair such that and , and

  3. for all with it holds .

The width of a directed path-decomposition is

The directed path-width of , for short, is the smallest integer such that there is a directed path-decomposition for of width .

There are a number of results on algorithms for computing directed path-width. The directed path-width of a digraph can be computed in time by [48] and in time by [52]. This leads to XP-algorithms for directed path-width w.r.t. the standard parameter and implies that for each constant , it is decidable in polynomial time whether a given digraph has directed path-width at most . Further it is shown in [49] how to decide whether the directed path-width of an -semicomplete digraph is at most in time . Furthermore the directed path-width can be computed in time , where denotes the vertex cover number of the underlying undirected graph of , by [50]. For sequence digraphs with a given decomposition into sequence the directed path-width can be computed in time , where denotes the maximum sequence length [35]. Further the directed path-width (and also the directed tree-width) can be computed in linear time for directed co-graphs [34].

Example for digraphs of small directed path-width are given in Example 3.3, when considering the equivalent (cf. Lemma 5.9) notation of directed vertex separation number.

3.2 Directed vertex separation number

The vertex separation number (vsn) for undirected graphs was introduced in [51]. In [60] the directed vertex separation number for a digraph has been introduced as follows.

Definition 3.2 (directed vertex separation number, [60])

The directed vertex separation number of a digraph is defined as follows.

(1)

Since the converse digraph has the same path-width as its original graph, we obtain an equivalent definition, which will be useful later on.

(2)
Example 3.3 (directed vertex separation number)
  1. Every directed path has directed vertex separation number 0.

  2. The -power graph of a directed path has directed vertex separation number 0.

  3. Every directed cycle has directed vertex separation number 1.

  4. The bidirectional complete digraph and the complete biorientation of a star have directed vertex separation number .333We use the complete biorientations of the two forbidden minors for the set of all graphs of vertex separation number 1, see [47, Fig. 1].

  5. Every bidirectional complete digraph has directed vertex separation number .

3.3 Directed cut-width

The cut-width (cutw) of undirected graphs was introduced in [1]. The cut-width of digraphs was introduced by Chudnovsky, Fradkin, and Seymour in [12].

Definition 3.4 (directed cut-width, [12])

The directed cut-width of digraph is

(3)

For every optimal layout we obtain the same value when we consider the arcs backwards in the reverse ordering . Thus we obtain an equivalent definition, which will be useful later on.

(4)

Subexponential parameterized algorithms for computing the directed cut-width of semicomplete digraphs are given in [25].

Example 3.5 (directed cut-width)
  1. Every directed path has directed cut-width 0.

  2. The -power graph of a directed path has directed directed cut-width .

  3. Every directed cycle has directed cut-width 1.

  4. The bidirectional complete digraph has directed cut-width .

  5. Every bidirectional complete digraph has directed cut-width .

3.4 Directed linear NLC-width

The linear NLC-width (lnlcw) for undirected graphs was introduced in [37] as a parameter by restricting the NLC-width444The abbreviation NLC results from the node label controlled embedding mechanism originally defined for graph grammars., defined in [59], to an underlying path-structure. Next we introduce the corresponding parameter for directed graphs by a modification of the edge inserting operation of the linear NLC-width, which also leads to a restriction of directed NLC-width [38]. Let be the set of all integers between and .

Definition 3.6 (directed linear NLC-width)

The directed linear NLC-width of a digraph , for short, is the minimum number of labels needed to define using the following four operations:

  1. Creation of a new vertex with label (denoted by ).

  2. Disjoint union of a labeled digraph and a single vertex labeled by plus all arcs between label pairs from directed from to and all arcs between label pairs from directed from to for two relations and (denoted by ).

  3. Change every label into label by some function (denoted by ).

The directed linear NLC-width of an unlabeled digraph is the smallest integer , such that there is a mapping such that the labeled digraph has directed linear NLC-width at most . An expression built with the operations defined above is called a directed linear NLC-width -expression. Note that every expression defines a layout by the order in which the vertices are inserted in the corresponding digraph. The digraph defined by expression is denoted by .

Example 3.7 (directed linear NLC-width)
  1. Every bidirectional complete digraph has directed linear NLC-width 1.

  2. The directed paths and have directed linear NLC-width 2.

  3. Every directed path has directed linear NLC-width at most 3.

  4. Every directed cycle has directed linear NLC-width at most 4.

  5. Every -power graph of a directed path has directed linear NLC-width at most .

  6. Every complete biorientation of a grid , , has directed linear NLC-width at least and at most , see [30, 33].

3.5 Directed linear clique-width

The linear clique-width (lcw) for undirected graphs was introduced in [37] as a parameter by restricting the clique-width, defined in [17], to an underlying path-structure. Next we introduce the corresponding parameter for directed graphs by a modification of the edge inserting operation of the linear clique-width, which also leads to a restriction for directed clique-width [17].

Definition 3.8 (directed linear clique-width)

The directed linear clique-width of a digraph , for short, is the minimum number of labels needed to define using the following four operations:

  1. Creation of a new vertex with label (denoted by ).

  2. Disjoint union of a labeled digraph and a single vertex labeled by (denoted by ).

  3. Inserting an arc from every vertex with label to every vertex with label (, denoted by ).

  4. Change label into label (denoted by ).

The linear clique-width of an unlabeled digraph is the smallest integer , such that there is a mapping such that the labeled digraph has linear linear clique-width at most . An expression built with the operations defined above is called a directed linear clique-width -expression. Note that every expression defines a layout by the order in which the vertices are inserted in the corresponding digraph. The digraph defined by expression is denoted by .

Example 3.9 (directed linear clique-width)
  1. Every edgeless digraph has directed linear clique-width 1.

  2. Every bidirectional complete digraph has directed linear clique-width 2.

  3. Every directed path has directed linear clique-width at most 3.

  4. Every directed cycle has directed linear clique-width at most 4.

  5. Every -power graph of a directed path has directed linear clique-width at most . For the given bound on the directed linear clique-width is even exact by Corollary 4.3.

  6. Every complete biorientation of a grid , , has directed linear clique-width at least and at most , see [30, 33].

3.6 Directed neighbourhood-width

The neighborhood-width (nw) for undirected graphs was introduced in [32]. It differs from linear NLC-width and linear clique-width at most by one but it is independent of vertex labels.

Let be a digraph and two disjoint vertex sets. The set of all out-neighbours of into set and the set of all in-neighbours of into set are defined by and . The directed neighbourhood of vertex into set is defined by and the set of all directed neighbourhoods of the vertices of set into set is . For some layout we define .

Definition 3.10 (directed neighbourhood-width)

The directed neighbourhood-width of a digraph is

Example 3.11 (directed neighbourhood-width)
  1. Every bidirectional complete digraph has directed neighbourhood-width 1.

  2. Every directed path has directed neighbourhood-width at most 2.

  3. Every directed cycle has directed neighbourhood-width at most 3.

  4. Every -power graph of a directed path has directed neighbourhood-width at most . For the given bound on the directed neighbourhood-width is even exact by Corollary 4.3.

  5. Every complete biorientation of a grid , , has directed neighbourhood-width at least and at most , see [30, 33].

3.7 Directed linear rank-width

The rank-width for directed graphs was introduced in Kanté in [44]. In [26] the linear rank-width (lrw) for undirected graphs was introduced by restricting the tree-structure of a rank decomposition to caterpillars, which is also possible for the directed case as follows.

Let a digraph and be a disjoint partition of the vertex set of . Further let be the adjacent matrix defined over the four-element field GF(4) for partition , i.e.

In GF(4) we have four elements with the properties and .

Definition 3.12 (directed linear rank-width)

A directed linear rank decomposition of digraph is a pair , where is a caterpillar (i.e. a path with pendant vertices) and is a bijection between and the leaves of . Each edge of divides the vertex set of by into two disjoint sets . For an edge in we define the width of as , i.e. the matrix rank555We denote by the rank of some matrix over , i.e. the number of independent lines or rows of . A set of rows

(i.e. vectors) are

independent, if there is no linear combination of a subset of to define a row in . A linear combination for some -tuple is for . of . The width of a directed linear rank decomposition is the maximal width of all edges in . The directed linear rank-width of a digraph , for short, is the minimum width of all directed linear rank decompositions for .

Example 3.13 (directed linear rank-width)
  1. Every bidirectional complete digraph and every directed path has directed linear rank-width 1.

  2. Every directed cycle has directed linear rank-width at most 2.

  3. Every complete biorientation of a grid , , has directed linear rank-width at least and at most , see [42, 33].

4 Directed width and undirected width

Next we compare the directed width of a digraph and the undirected width of its underlying undirected graph .

Theorem 4.1

Let be a directed graph.

Proof 

  1. A path-decomposition for of width is also a directed path-decomposition for of width .

  2. Let be a digraph and be the underlying undirected graph of cut-width . Let be the corresponding ordering of the vertices, such that for every , there are at most edges such that and . Since every undirected edge in comes from a directed edge , a directed edge , or both, and the directed cut-width only counts edges directed forward, the same layout shows that the directed cut-width of is at most .

  3. Let be a digraph of directed neighbourhood-width and a linear layout, such that for every it holds . Since for every pair of vertices in of the same directed neighbourhood the corresponding vertices in have the same neighbourhood, it follows that for every it holds . Thus, the neighbourhood-width of is at most .

    Let be a digraph and be the underlying undirected graph of neighbourhood-width . Then there is a layout , such that for every the vertices in can be divided into at most subsets , such that the vertices of set , , have the same neighbourhood with respect to the vertices in . One of these sets may consist of vertices having no neighbors . Every of the remaining sets has at most vertices such that there is an edge with . Let .

    • If there is one set which consists of vertices having no neighbours , then there are at most vertices , such that there is an edge with .

    • Otherwise there are at most vertices , such that there is an edge with .

    Thus for every the vertices in can be divided into subsets , such that the vertices of set , , have the same directed neighbourhood with respect to the vertices in . Thus the directed neighbourhood-width of is at most .

  4. Let be a digraph of directed linear NLC-width and be a directed linear NLC-width -expression for . A linear NLC-width -expression for can recursively be defined as follows.

    • Let for . Then .

    • Let for . Then .

    • Let for and . Then .

    The second bound follows by

  5. Let be a digraph of directed linear clique-width and be a directed linear clique-width -expression for . A linear clique-width -expression for can recursively be defined as follows.

    • Let for . Then .

    • Let for . Then .

    • Let for . Then .

    • Let for . Then .

    The second bound follows by

  6. Let be a digraph of directed linear rank-width and be a directed linear rank-decomposition for of width . Then is also a linear rank-decomposition for . Let be an edge of . Let be the adjacent matrix defined over the two-element field GF(2) for partition . If for two rows in are linearly dependent then for these two rows in are also linearly dependent. Thus we conclude that and thus linear rank-width of .

    The second bound follows by

This completes the proof.

Remark 4.2

In Theorem 4.1(1) and (2) the directed path-width of some digraph can not be used to give an upper bound on the path-width of . Any transitive tournament has directed path-width but its underlying undirected graph has a path-width which corresponds to the number of vertices. Also by restricting the vertex degree this is not possible by an acyclic orientation of a grid. The same examples also show that the directed cut-width of some digraph can not be used to give an upper bound on the cut-width of .

The relations shown in Theorem 4.1 allow to imply the following values for the directed linear clique-width and directed neighbourhood-width of a -power graph of a path.

Corollary 4.3
  1. For it holds .

  2. For it holds .

Proof  For we know from [40] that the (undirected) linear clique-width of a -power graph of a path on vertices is exactly .

  1. For by

    it holds .

  2. For by

    it holds .

This completes the proof.

Comparing the undirected width of a graph and the directed width of its complete biorientation the following results hold.

Theorem 4.4

For each width measure and every undirected graph it holds .

Proof 

  1. Since is the underlying undirected graph of , by Theorem 4.1(1.) it remains to show that the path-width of is at most the directed path-width of . Let be a directed path-decomposition for . For every it holds and for . If then since in there is also the arc we obtain a contradiction. Thus it holds which implies that the given path-decomposition is also a path-decomposition for .

  2. By Theorem 4.1(2.) it remains to show that the cut-width of is at most the directed cut-width of . Let be a graph and its complete biorientation of directed cut-width . Let be the corresponding ordering of the vertices, such that for every , there are at most arcs such that and . Since every such arc corresponds to one undirected edge in , the same layout shows that the cut-width of is at most .

  3. By Theorem 4.1(3.) it remains to show that the directed neighbourhood-width of is at most the neighbourhood-width of . Let a linear layout, such that for every for the number of neighbourhoods it holds . By the definitions of and for neighbourhoods of directed graphs, it follows that for every for the number of directed neighbourhoods it holds