1 Introduction
A graph parameter is a function that associates with every graph a positive integer. Examples for graph parameters are treewidth [57], cliquewidth [17], NLCwidth [59], and rankwidth [55]. Cliquewidth, NLCwidth, and rankwidth are equivalent, i.e. a graph has bounded cliquewidth, if and only if it has bounded NLCwidth and that is if and only if it has bounded rankwidth. The latter three parameters are more general than treewidth, since graphs of bounded treewidth also have bounded cliquewidth but even for dense graphs (e.g. cliques) the treewidth is unbounded while the cliquewidth 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 pathstructure for the input graph. These are pathwidth [56], cutwidth [1], linear cliquewidth [37], linear NLCwidth [37], neighbourhoodwidth [32], and linear rankwidth [26]. With the exception of cutwidth these parameters can be regarded as restrictions of the above mentioned parameters with underlying treestructure to an underlying pathstructure. The relation between these parameters corresponds to their treestructural counterparts, since bounded pathwidth implies bounded linear NLCwidth, linear cliquewidth, neighborhoodwidth, and linear rankwidth. Further the reverse direction is not true in general, see [32]. Such restrictions to underlying pathstructures 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 treestructure to directed graphs lead to directed treewidth [43], directed NLCwidth [38], directed cliquewidth [17], and directed rankwidth [44].
In this paper we study directed graph parameters which are defined by the existence of an underlying pathstructure for the input graph. One of the most famous examples is the directed pathwidth, which was introduced by Reed, Seymour, and Thomas around 1995 (see [7]) and studied in [7, 58, 49, 48]. Further the cutwidth 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 NLCwidth, directed linear cliquewidth, directed neighborhoodwidth, and directed linear rankwidth. 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  

cutwidth  cutw  [1]  dcutw  [12]  0  0  0  
pathwidth  pw  [56]  dpw  Thomas et al.  0  1  0  0  
linear cliquewidth  lcw  [37]  dlcw  here  2  2  3  2  
linear NLCwidth  lnlcw  [37]  dlnlcw  here  1  1  3  1  
neighbourhoodwidth  nw  [32]  dnw  here  1  1  2  1  
linear rankwidth  lrw  [26]  dlrw  here  1  1  2  1 
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 rankwidth, 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 BangJensen 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
^{1}^{1}1Thus we do not consider directed graphs with loops. vertices called arcs. For a vertex , the sets and are called the set of all outneighbours and the set of all inneighbours of . The outdegree of , for short, is the number of outneighbours of and the indegree of , for short, is the number of inneighbours of in . The maximum outdegree is and the maximum indegree 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 outtree (intree) 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 pathwidth
The pathwidth (pw) for undirected graphs was introduced in [56]. The notion of directed pathwidth was introduced by Reed, Seymour, and Thomas around 1995 (cf. [7]) and relates to directed treewidth introduced by Johnson, Robertson, Seymour, and Thomas in [43].^{2}^{2}2Please note that there are some works which define the pathwidth of a digraph in a different and not equivalent way by using the pathwidth of , see Section 7.
Definition 3.1 (directed pathwidth)
Let be a digraph. A directed pathdecomposition of is a sequence of subsets of , called bags, such that the following three conditions hold true.

,

for each there is a pair such that and , and

for all with it holds .
The width of a directed pathdecomposition is
The directed pathwidth of , for short, is the smallest integer such that there is a directed pathdecomposition for of width .
There are a number of results on algorithms for computing directed pathwidth. The directed pathwidth of a digraph can be computed in time by [48] and in time by [52]. This leads to XPalgorithms for directed pathwidth w.r.t. the standard parameter and implies that for each constant , it is decidable in polynomial time whether a given digraph has directed pathwidth at most . Further it is shown in [49] how to decide whether the directed pathwidth of an semicomplete digraph is at most in time . Furthermore the directed pathwidth 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 pathwidth can be computed in time , where denotes the maximum sequence length [35]. Further the directed pathwidth (and also the directed treewidth) can be computed in linear time for directed cographs [34].
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 pathwidth as its original graph, we obtain an equivalent definition, which will be useful later on.
(2) 
Example 3.3 (directed vertex separation number)

Every directed path has directed vertex separation number 0.

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

Every directed cycle has directed vertex separation number 1.

The bidirectional complete digraph and the complete biorientation of a star have directed vertex separation number .^{3}^{3}3We use the complete biorientations of the two forbidden minors for the set of all graphs of vertex separation number 1, see [47, Fig. 1].

Every bidirectional complete digraph has directed vertex separation number .
3.3 Directed cutwidth
The cutwidth (cutw) of undirected graphs was introduced in [1]. The cutwidth of digraphs was introduced by Chudnovsky, Fradkin, and Seymour in [12].
Definition 3.4 (directed cutwidth, [12])
The directed cutwidth 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 cutwidth of semicomplete digraphs are given in [25].
Example 3.5 (directed cutwidth)

Every directed path has directed cutwidth 0.

The power graph of a directed path has directed directed cutwidth .

Every directed cycle has directed cutwidth 1.

The bidirectional complete digraph has directed cutwidth .

Every bidirectional complete digraph has directed cutwidth .
3.4 Directed linear NLCwidth
The linear NLCwidth (lnlcw) for undirected graphs was introduced in [37] as a parameter by restricting the NLCwidth^{4}^{4}4The abbreviation NLC results from the node label controlled embedding mechanism originally defined for graph grammars., defined in [59], to an underlying pathstructure. Next we introduce the corresponding parameter for directed graphs by a modification of the edge inserting operation of the linear NLCwidth, which also leads to a restriction of directed NLCwidth [38]. Let be the set of all integers between and .
Definition 3.6 (directed linear NLCwidth)
The directed linear NLCwidth of a digraph , for short, is the minimum number of labels needed to define using the following four operations:

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

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 ).

Change every label into label by some function (denoted by ).
The directed linear NLCwidth of an unlabeled digraph is the smallest integer , such that there is a mapping such that the labeled digraph has directed linear NLCwidth at most . An expression built with the operations defined above is called a directed linear NLCwidth 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 NLCwidth)

Every bidirectional complete digraph has directed linear NLCwidth 1.

The directed paths and have directed linear NLCwidth 2.

Every directed path has directed linear NLCwidth at most 3.

Every directed cycle has directed linear NLCwidth at most 4.

Every power graph of a directed path has directed linear NLCwidth at most .
3.5 Directed linear cliquewidth
The linear cliquewidth (lcw) for undirected graphs was introduced in [37] as a parameter by restricting the cliquewidth, defined in [17], to an underlying pathstructure. Next we introduce the corresponding parameter for directed graphs by a modification of the edge inserting operation of the linear cliquewidth, which also leads to a restriction for directed cliquewidth [17].
Definition 3.8 (directed linear cliquewidth)
The directed linear cliquewidth of a digraph , for short, is the minimum number of labels needed to define using the following four operations:

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

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

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

Change label into label (denoted by ).
The linear cliquewidth of an unlabeled digraph is the smallest integer , such that there is a mapping such that the labeled digraph has linear linear cliquewidth at most . An expression built with the operations defined above is called a directed linear cliquewidth 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 cliquewidth)

Every edgeless digraph has directed linear cliquewidth 1.

Every bidirectional complete digraph has directed linear cliquewidth 2.

Every directed path has directed linear cliquewidth at most 3.

Every directed cycle has directed linear cliquewidth at most 4.

Every power graph of a directed path has directed linear cliquewidth at most . For the given bound on the directed linear cliquewidth is even exact by Corollary 4.3.
3.6 Directed neighbourhoodwidth
The neighborhoodwidth (nw) for undirected graphs was introduced in [32]. It differs from linear NLCwidth and linear cliquewidth at most by one but it is independent of vertex labels.
Let be a digraph and two disjoint vertex sets. The set of all outneighbours of into set and the set of all inneighbours 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 neighbourhoodwidth)
The directed neighbourhoodwidth of a digraph is
Example 3.11 (directed neighbourhoodwidth)

Every bidirectional complete digraph has directed neighbourhoodwidth 1.

Every directed path has directed neighbourhoodwidth at most 2.

Every directed cycle has directed neighbourhoodwidth at most 3.

Every power graph of a directed path has directed neighbourhoodwidth at most . For the given bound on the directed neighbourhoodwidth is even exact by Corollary 4.3.
3.7 Directed linear rankwidth
The rankwidth for directed graphs was introduced in Kanté in [44]. In [26] the linear rankwidth (lrw) for undirected graphs was introduced by restricting the treestructure 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 fourelement field GF(4) for partition , i.e.
In GF(4) we have four elements with the properties and .
Definition 3.12 (directed linear rankwidth)
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 rank^{5}^{5}5We 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 rankwidth of a digraph , for short, is the minimum width of all directed linear rank decompositions for .Example 3.13 (directed linear rankwidth)
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

A pathdecomposition for of width is also a directed pathdecomposition for of width .

Let be a digraph and be the underlying undirected graph of cutwidth . 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 cutwidth only counts edges directed forward, the same layout shows that the directed cutwidth of is at most .

Let be a digraph of directed neighbourhoodwidth 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 neighbourhoodwidth of is at most .
Let be a digraph and be the underlying undirected graph of neighbourhoodwidth . 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 neighbourhoodwidth of is at most .


Let be a digraph of directed linear NLCwidth and be a directed linear NLCwidth expression for . A linear NLCwidth expression for can recursively be defined as follows.

Let for . Then .

Let for . Then .

Let for and . Then .
The second bound follows by


Let be a digraph of directed linear cliquewidth and be a directed linear cliquewidth expression for . A linear cliquewidth 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


Let be a digraph of directed linear rankwidth and be a directed linear rankdecomposition for of width . Then is also a linear rankdecomposition for . Let be an edge of . Let be the adjacent matrix defined over the twoelement 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 rankwidth of .
The second bound follows by
This completes the proof.
Remark 4.2
In Theorem 4.1(1) and (2) the directed pathwidth of some digraph can not be used to give an upper bound on the pathwidth of . Any transitive tournament has directed pathwidth but its underlying undirected graph has a pathwidth 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 cutwidth of some digraph can not be used to give an upper bound on the cutwidth of .
The relations shown in Theorem 4.1 allow to imply the following values for the directed linear cliquewidth and directed neighbourhoodwidth of a power graph of a path.
Corollary 4.3

For it holds .

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

For by
it holds .

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

Since is the underlying undirected graph of , by Theorem 4.1(1.) it remains to show that the pathwidth of is at most the directed pathwidth of . Let be a directed pathdecomposition 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 pathdecomposition is also a pathdecomposition for .

By Theorem 4.1(2.) it remains to show that the cutwidth of is at most the directed cutwidth of . Let be a graph and its complete biorientation of directed cutwidth . 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 cutwidth of is at most .

By Theorem 4.1(3.) it remains to show that the directed neighbourhoodwidth of is at most the neighbourhoodwidth 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
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