A digraph is reflexive if each ( has all loops); irreflexive if no ( has no loops); and symmetric if implies . In this paper, we shall treat both graphs and digraphs; for simplicity we view graphs as symmetric digraphs. (Thus, graphs can have loops, and irreflexive graphs are loopless.) Loops play an important role in this paper, and this is not common in the literature on graph classes that we consider. They allow us to view several seemingly unrelated graph classes through a common lens.
A min ordering of a digraph is a linear ordering of the vertices of , so that and implies that . In other words, a min ordering is an ordering of the vertices such that when the rows and columns of the adjacency matrix are ordered in this way, neither the matrix whose rows are and nor the matrix whose rows are and appears as a submatrix. (See Figure 1.) Note that the presence or absence of loops (1’s on the diagonal of the adjacency matrix) can affect whether the graph has a min ordering.
Our goal in this paper is to promote a class of digraphs (or 0,1-matrices) that is a broad generalization of interval graphs and that retains some of the desirable structural properties of interval graphs. A graph is an interval graph if it is the intersection graph of a family of intervals on the real line, i.e., if there exists a family of intervals such that if and only if . The family of intervals is an interval model of . (See Figure 2.) We note that the definition implies that an interval graph is reflexive. A related concept for bipartite graphs is as follows. A bipartite graph with parts is an interval bigraph if there are intervals , and , such that for and , if and only if .
Interval graphs are important in graph theory and in applications, and are distinguished by several elegant characterizations and efficient recognition algorithms [3, 9, 13, 15, 18, 29, 36]. For this reason, there have been attempts to extend the concept to digraphs , with mixed success. (Many of the desirable structural properties are absent.) More recently a more restricted class of digraphs has been found to offer a nicer generalization of interval graphs; these are the adjusted interval digraphs . A digraph is an adjusted interval digraph if there are two families of real intervals, the source intervals and the sink intervals and such that if and only if the source interval for intersects the sink interval for . (See Figure 3.) This differs from the class in  in that the left endpoint, , must be shared by the two intervals and assigned to ; they are “adjusted.” The interval graphs are the special case where for each . An adjusted interval model of is a set of source and sink intervals that represent in this way.
Adjacency on a set of intervals can also be defined by interval containment. A graph is a containment graph of intervals  if there is a family of intervals on the real line such that if and only if one of and contains the other. A graph is a containment graph of intervals if and only if it and its complement are both transitively orientable, thus if and only if it is a permutation graph .
For this paper, a more relevant class is a bipartite version of this concept. A bipartite graph with parts is an interval containment bigraph  if there are sets of intervals , and }, such that if and only if . These graphs have been studied, from the point of view of another geometric representation, as two-directional orthogonal ray graphs . A bipartite graph with parts and is called a two-directional orthogonal ray graph if there exists a set of upwards vertical rays, and a set of horizontal rays to the right such that if and only if . It is known that a bipartite graph is an interval containment graph if and only if it is a two-directional orthogonal ray graph , and if and only if its complement is a circular arc graph .
It is sometimes convenient to view bipartite graphs as digraphs, with all edges oriented from part to part ; thus we speak of a bipartite interval containment digraph, a bipartite interval digraph, or a two-directional orthogonal ray digraph. In general, a bipartite digraph is a bipartite graph with parts and and all arcs being oriented from to .
There is an interesting intermediate concept that uses both intersection and containment of intervals to define adjacency. An interval model of an interval graph can be viewed as two mappings and such that for each , and such that if and only if and . The constraint comes from the need for to be an interval. The proposition that two intervals intersect is the same as and , since this means that neither interval lies entirely to the right of the other.
A generalization of interval models is obtained by dropping the constraint in this formulation. To develop the motivation for this, we start with the complements of threshold tolerance graphs. A graph is a threshold tolerance graph  if its vertices can be assigned weights and tolerances so that is an edge of if and only if or . (When all are equal, this defines a better known class of threshold graphs .) Co-threshold tolerance (‘co-TT’) graphs are complements of threshold tolerance graphs. Equivalently, a graph is a co-TT graph, if there exist real numbers , such that if and only if and . This differs from the definition of interval graphs in that it is no longer required that , illustrating the motivation for dropping the constraint in this case. (See Figure 4.) That these are precisely the co-TT graphs is easily seen by letting and . The two mappings and , are called the co-TT model of .
One view of a co-TT model is that there are now intervals whose ‘beginning’ may come after their ‘end’ . In other words, we may have ‘intervals’ with . We may view a co-TT model as consisting of intervals some of which go in the positive direction (have ) and others go in the negative direction (have ). We speak of positive or negative intervals, and positive or negative vertices that correspond to them. (In the literature [14, 16, 20, 21, 32], the direction is denoted by colors of the intervals: positive intervals, and vertices, are colored blue, and negative intervals, and vertices, are colored red.) The above definition of adjacency has interesting consequences. Two positive vertices are adjacent if and only if they intersect; in particular, each positive vertex has a loop. Two negative vertices are never adjacent; in particular negative vertices have no loops. Finally, a positive vertex corresponding to a positive interval and a negative vertex corresponding to a negative interval are adjacent if and only if is contained in (i.e., ). We also use the following signed shorthand, which will be useful later: a positive vertex or interval will be called a -vertex or -interval respectively, and a negative vertex or interval will be called a -vertex or -interval respectively. It follows from the above discussion that in a co-TT graph, the -vertices induce a reflexive interval graph, the -vertices form an independent set, and the edges between the -vertices and the -vertices form a bipartite interval containment graph.
Note that co-TT graphs are a generalization of interval graphs; the interval graphs are those co-TT graphs where all vertices are positive. In other words, they are the reflexive co-TT graphs.
2 Signed Interval Digraphs
We have now seen extensions of interval graphs in two different directions. First, by taking two (adjusted) intervals instead of just one interval, we were able to extend the definition from reflexive graphs to reflexive digraphs. Second, by admitting intervals that go in the negative direction (have ), we were able to extend the definition from reflexive graphs to graphs that have some vertices with loops and others without. Both these generalizations have proved very fruitful [12, 9, 12, 14, 26, 16, 20, 21, 32].
We now define a new class of digraphs that unifies these extensions. A digraph is a signed-interval digraph if there exist three mappings from to the real line, , and , such that if and only if and . We call the three mappings , and a signed-interval model of . Alternatively, a signed interval model is obtained in by assigning, for each a source interval and a sink interval , such that if and only if and . (See figure 5.) Since it is possible that and/or , each of and can be negative or positive. Since the source interval and sink interval for share the endpoint , we retain the property that the intervals are adjusted.
Signed-interval digraphs with all intervals positive, are reflexive, and are adjusted interval digraphs. Signed-interval digraphs with , for all , are symmetric, and are co-TT graphs. Signed-interval digraphs that satisfy both conditions, i.e., with all , are interval graphs. Furthermore, we show below that there are no reflexive signed-interval digraphs other than adjusted interval digraphs, no symmetric signed-interval digraphs other than co-TT graphs, and no reflexive and symmetric signed-interval digraphs other than interval graphs.
The structure of signed-interval digraphs can be described in a language similar to what was used for co-TT graphs. Let be a signed-interval digraph and consider a signed-interval model of
given by the ordered pairsof intervals where and . For , we say a vertex is of type if is an -interval and is a -interval. The subdigraph of induced by -vertices is an adjusted interval digraph. The -vertices of form an independent set. The arcs between the - and -vertices form a bipartite interval containment digraph. The arcs between the - and -vertices also form a bipartite interval containment digraph. Similar properties hold for the other parts and their connections.
We emphasize that our definition of co-TT graphs differs from the standard definition [14, 16, 32]. In the standard definition, the condition and is applied only for , and so the graphs have no loops. Thus a graph under the standard interpretation is co-TT if and only if with a suitable addition of loops it is co-TT under our definition above. This difference is not important as it was shown in  that if a graph is co-TT (in the standard sense), then it has a co-TT model with negative intervals for all simplicial vertices without true twins and all other intervals positive. Thus there is an easy translation between the co-TT graphs as defined here and the standard irreflexive co-TT graphs: namely, loops are to be placed on all vertices other than simplicial vertices without true twins.
3 Min Orderings
Interval graphs, adjusted interval digraphs, co-TT graphs, and two-directional orthogonal ray digraphs all have min orderings when care is taken to specify which vertices have loops and which do not. [9, 12, 23, 34].
Min orderings are a useful tool for graph homomorphism problems. A homomorphism of a digraph to a digraph is a mapping such that whenever . If a digraph has a min ordering, there is a simple polynomial-time algorithm to decide if a given input graph admits a homomorphism to a fixed digraph [17, 24]. In fact, the algorithm is well known in the AI community as the arc-consistency algorithm ; it is easy to see that it also solves list homomorphism problems, where we seek a homomorphism of input to fixed taking each vertex of to one of a ‘list’ of allowed images. In fact, many (but not all) homomorphism and list homomorphism problems that can be solved in polynomial time can be solved using arc-consistency with respect to a min ordering.
Graph and digraph homomorphism problems are special cases of constraint satisfaction problems. A general tool for solving polynomial time solvable constraint satisfaction problems are the so-called polymorphisms . Without going into the technical details, we mention that min-orderings are equivalent to conservative semilattice polymorphisms .
We prove below that a digraph has a min ordering if and only if it is a signed-interval digraph. We also give another geometric characterization of signed-interval digraphs, as bi-arc digraphs. We show that a reflexive signed-interval digraphs are precisely adjusted interval digraphs, that symmetric signed-interval digraphs are precisely co-TT graphs, that reflexive and symmetric signed-interval digraphs are precisely interval graphs, and that bipartite signed-interval digraphs are precisely two-directional ray graphs.
The main result of this section is the following.
A digraph admits a min ordering if and only if it is a signed-interval digraph.
Before embarking on the proof we offer an alternate definition of a min ordering. Consider any linear ordering of . To this ordering, we prepend an intial element , which is a place holder and not a vertex. Thus, for each vertex . We denote by the last vertex (in the order ), such that is an out-neighbor of (i.e., such that ), or if has no out-neighbor. Similarly, for each vertex , we denote by the last vertex such that is an in-neighbor of (i.e., such that ), or if has no in-neighbor.
A linear ordering of is a min ordering of a digraph if and only if the following property holds:
if and only if and .
Suppose first that is a min ordering of with prepended. If , then by the definition of we have and . On the other hand, let and . Note that if or we have also by definition. Therefore it remains to consider vertices such that and . Then and the min ordering property implies that . This proves the property.
Conversely, assume that is a linear ordering of with prepended and that the property holds for . We claim it is a min ordering of . Otherwise some would have . This is a contradiction, since we have and . ∎
We proceed to prove the theorem.
Suppose is a min ordering of a digraph with prepended. We represent each vertex by the mappings . In other words, is represented by the two intervals and . It follows from Proposition 3 that if and only if and . Thus is a signed-interval digraph.
Conversely, suppose we have the three mappings from to the real line, such that if and only if and . Without loss of generality we may assume the points are all distinct. Then we claim that the left to right ordering of the points yields a min ordering of . (Specifically, we define if and only if precedes .) Consider now with . This means that and , whence we must have . ∎
4 An alternate geometric representation
Digraphs that admit a min ordering have another geometric representation. Let be a circle with two distinguished points (the poles) and , and let be a digraph. Let and be two families of arcs on such that each contains but not , and each contains but not . We say that the families and are consistent if they have the same clockwise order of their clockwise ends, i.e., the clockwise end of precedes in the clockwise order the clockwise end of if and only if the clockwise end of precedes in the clockwise order the clockwise end of . Suppose two families are consistent; we define an ordering on where if and only if the clockwise end of precedes in the clockwise order the clockwise end of ; we call the ordering generated by the consistent families .
A bi-arc model of a digraph is a consistent pair of families of circular arcs, , such that if and only if and are disjoint. A digraph is called a bi-arc digraph if it has a bi-arc model.
A digraph admits a min ordering if and only if it is a bi-arc digraph.
Suppose form a bi-arc model of . We claim that the ordering generated by is a min ordering of . Indeed, suppose and have . Then spans the area of the circle between and the clockwise end of , and spans the area of the circle between and the clockwise end of . (See Figure 1.) This implies that and are disjoint: indeed, the counterclockwise end of is blocked from reaching by (since ), and the counterclockwise end of is blocked from reaching by (since ). (The clockwise ends are fixed by the ordering .)
Conversely, suppose is a min ordering of . We construct families of arcs and , with , as follows. The intervals will contain but not , the intervals will contain but not . The clockwise ends of are arranged in clockwise order according to , as are the clockwise ends of . The counterclockwise ends will now be organized so that , becomes a bi-arc model of . For each vertex , we define and as in the proof of Theorem 1. Then we assign the counterclockwise endpoint of to be N if has no out-neighbors, or else extend counterclockwise as far as possible without intersecting , and assign the the counterclockwise endpoint of each to be if has no in-neighbors, or else extend counterclockwise as far as possible without intersecting . We claim this is a bi-arc model of . Clearly, if , then intersects by the construction, and similarly for we have intersecting . This leaves disjoint all pairs such that and ; since , the definition of min ordering implies that , as required. ∎
The following statements are equivalent for a digraph .
has a min ordering
is a signed-interval digraph
is a bi-arc digraph.
5 0,1-Matrices and bipartite graphs
Irreflexive graphs with at least one edge do not admit a min ordering, since the vertices that are not reflexive form an independent set. However, in the special case of bipartite graphs, a version of min ordering has been studied, and has yielded interesting examples. We will describe that version below, but we first want to explain how to view that definition as a special case of min ordering as defined here.
A useful perspective on min orderings is obtained by considering 0,1-matrices. Square 0,1-matrices naturally correspond to adjacency matrices of digraphs. Let a simultaneous permutation of rows and columns of a matrix be one where the permutation of the rows is the same as the permutation of the columns. An independent permutation of rows and columns allows the permutation of the rows to be different from the permutation of the columns.
Let be the two by two matrix with rows and , and let be the two by two matrix with rows and
. (These have been given other names in the literature, up to a simultaneous permutation of rows and columns they are the gamma matrix, and the identity matrix.) A matrixis called -free if it does not contain or as a submatrix. If is the adjacency matrix of a digraph , and if the rows and columns of are in the order , then is -free if and only if is a min ordering. We call a a min-orderable matrix if its rows and columns can be simultaneously permuted to produce a -free matrix. A digraph has a min ordering if and only if its adjacency matrix is min-orderable.
Therefore, we can say much about matrices that are min-orderable.
A square 0,1-matrix is min-orderable if and only if it is the adjacency matrix of a signed-interval digraph.
Another natural interpretation of a 0,1-matrix is that it represents adjacencies in a bipartite graph, with rows corresponding to one part and columns to the other part. The bi-adjacency matrix of a bipartite graph with marts has its -th entry is if and only if the -th vertex in is adjacent to the -th vertex in . Note that for this interpretation it is not required that the matrix be square. For matrices that are not necessarily square, we can still ask for independent permutations of rows and columns, to produce a -free matrix. This suggests a definition of min ordering for bipartite graphs as follows. A min ordering of a bipartite graph with parts and is a linear ordering of and a linear ordering of so that for any such that and we have . This is the definition that has been used in the literature; it is clear how it avoids the problems of the general definition.
There is a simple transformation that connects the two interpretations of 0,1-matrices. For a matrix with rows and columns, we define the by square matrix to contain the matrix in the first rows and the last columns, with everywhere else. Then a simultaneous row/column permutation of corresponds to independent row and column permutations of . Note that the square matrix is an adjacency matrix of the digraph obtained from by directing all edges from from the first part to the second part. Thus to view the special definition of a min ordering for bipartite graphs as a particular case of the general definition, it suffices to view bipartite graphs as digraphs with all edges oriented from the first part to the second part. We shall say that is a bipartite digraph if it is obtained from a bipartite graph in this way.
A robust class of bipartite graphs is relevant for our discussion. A bipartite graph with parts and is called a two-directional orthogonal ray graph if there exists a set of upwards vertical rays, and a set of horizontal rays to the right such that if and only if . Note that we may, if needed, view a two-directional orthogonal ray graph as a bipartite digraph, with all edges oriented from (say) vertical rays to horizontal rays.
The following statements are equivalent for a bipartite graph .
is a two-directional orthogonal ray graph
the complement of is a circular arc graph
is an interval containment graph.
Matrices that can be permuted to avoid small submatrices have been of much interest [1, 28, 30]. This of course corresponds to characterizations of digraphs by forbidden ordered subgraphs [7, 22]. Our focus was on -free matrices. Let the matrix be obtained from by simultaneously exchanging the rows and columns; i.e., has rows , . Let be the two by two identity matrix. It is easy to see that considering -free matrices is equivalent to considering -free matrices, as the permutation that simultaneously reverses rows and columns of matrix transforms a -free matrix to a -free matrix and vice versa. Matrices that are -free have been intensively studied [1, 30], cf. . A bipartite graph is chordal bipartite if it contains no induced cycle other than . A reflexive graph is strongly chordal if it contains no induced cycle or induced trampoline. (A trampoline is a complete graph on with vertices each only adjacent to , subscripts computed modulo .) The adjacency matrix of a reflexive graph can be made -free by simultaneous row / column permutations if and only if is strongly chordal; the bi-adjacency matrix of a bipartite graph can be made -free by independent permutations of rows and columns if and only if is chordal bipartite . These results amount to forbidden structure characterizations of matrices that are permutable (by simultaneous or independent row and column permutations) to a -free format. Algorithms to recognize such matrices efficiently have been given in [30, 33]. For -free matrices, or equivalently, for -free matrices a forbidden structure characterization is given in . An recognition algorithm is claimed in , cf. .
6 Special cases
We now explore what min orderings look like in the special cases we have discussed, namely reflexive graphs, reflexive digraphs, undirected graphs, and bipartite graphs. The results are all corollaries of Theorem 3 and Proposition 3.
A reflexive graph is a signed-interval digraph if and only if it is an interval graph.
A reflexive digraph is a signed-interval digraph if and only if it is an adjusted interval digraph.
Next we focus on symmetric digraphs, i.e., graphs.
A graph is a signed-interval digraph, i.e., has a min ordering, if and only if it is a co-TT graph.
Consider a co-TT model of , given by the mappings , setting the third mapping with each , yields a signed-interval digraph model of . Conversely, assume is a graph, i.e., a symmetric digraph, that is a signed-interval digraph. Let be a min ordering of ; we again have for all vertices . We claim that the mappings define a co-TT model. Indeed, from Proposition 3 we have if and only if and , as required. ∎
Finally, for bipartite graphs we have the following result, stated for convenience in the language of bipartite digraphs. Note that this has only one consequence, that is, when we consider an edge we always assume and .
A bipartite digraph is a signed-interval digraph, i.e., has a min ordering, if and only if it is a two-directional orthogonal ray graph.
Note that two-directional orthogonal ray graphs themselves have two other equivalent characterizations in Theorem 5. The characterization of two-directional orthogonal ray graph by the existence of a min ordering is also observed in [23, 34].
Suppose has a signed-interval model given by the three mappings such that if and only if and . We construct a two-directional ray model for as follows. For each , we take an upwards vertical ray starting in the point with -coordinate equal to and with -coordinate equal to . For each , we take a horizontal ray to the right, starting in the point with -coordinate and -coordinate . Now intersects if and only if and , i.e., if and only if as required.
Now suppose that has a two-directional model, i.e., upwards vertical rays and horizontal rays to the right such that if and only if . We will prove that has a min ordering, whence it is a signed-interval digraph by Theorem 3. We will define the orders on and on as follows. Assume the starting point of the vertical ray has the -coordinates , and the starting point of the horizontal ray has the -coordinates , for and . It is easy to see that we may assume, without loss of generality, that all and are distinct, and similarly for and . We define in if and only if , and define in if and only if . We show that this is a min ordering of the bipartite digraph . Otherwise, some have . There are two possibilities for ; either or . In the former case, , in the latter case , contradicting the assumptions. ∎
7 Algorithms and characterizations
Interval graphs are known to have elegant characterization theorems [13, 29], cf. [15, 36] and efficient recognition algorithms [3, 5, 18]. Thus one might hope to be able to obtain similar results for their generalizations and digraph analogues. This is true for all the generalizations described in this paper, at least to some degree. In this section we summarize what is known.
The prototypical characterization of interval graphs is the theorem of Lekkerkerker and Boland . In our language, it states that a reflexive graph is an interval graph if and only if it contains no asteroidal triple and no induced or . An asteroidal triple consists of three non-adjacent vertices such that any two are joined by a path not containing any neighbors of the third vertex. An equivalent characterization by the absence of a slightly less concise obstruction is given in . A reflexive graph is an interval graph if and only if it contains no invertible pair. An invertible pair is a pair of vertices such that there exist two walks of equal length, from to , and from to , where the -th vertex of is non-adjacent to the -st vertex of (for each ), and also two walks of equal length from to and to respectively, where the -th vertex of is non-adjacent to the -st vertex of (for each ). It is not difficult to see that an asteroidal triple is a special case of an invertible pair. A number of variants of the definition of an invertible pair have arisen [12, 20, 21, 23], and they have proved useful to give characterization theorems for various classes. It is proved in  that a reflexive digraph is an adjusted interval digraph if and only if it contains no directed invertible pair. A directed version of an invertible pair is defined in  in a manner similar to the above definition of an invertible pair. With yet another labeled version of an invertible pair, we have the following obstruction characterization of co-TT graphs: a graph is a co-TT graph if and only if it contains no labeled invertible pair, which follows from the characterization in  in terms of an interval ordering from . For bipartite graphs, an analogous bipartite version of an invertible pair yields the following result. A bipartite graph is a two-directional orthogonal ray graph if and only if it contains no bipartite invertible pair, . In fact, in  a stronger version is shown: there is a bipartite analogue of an asteroidal triple, called an edge-asteroid, and a bipartite graph is a two-directional orthogonal ray graph if and only if it contains no induced -cycle and no edge-asteroid. Finally, in , there is an obstruction characterization for signed-interval digraphs, which is a little more technical than just an invertible pair, .
There is a long history of efficient algorithms for the recognition of interval graphs, many of them linear time, starting from  and culminating in . A polynomial time algorithm for the recognition of adjusted interval digraphs is given in . It is not known how to obtain a linear time, or even near-linear time algorithm. An algorithm for the recognition of two-directional orthogonal ray graphs follows from Theorem 5 and . A more efficient algorithm in this case is also not known. On the other hand, an algorithm for the recognition of co-TT graphs has been given in . In , a polynomial-time algorithm for the recognition of a signed-interval digraph is proposed. (A new version of  will be posted on arXiv soon.)
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