Threshold Digraphs

A digraph whose degree sequence has a unique vertex labeled realization is called threshold. In this paper we present several characterizations of threshold digraphs and their degree sequences, and show these characterizations to be equivalent. Using this result, we obtain a new, short proof of the Fulkerson-Chen theorem on degree sequences of general digraphs.


Introduction
Creating models of real-world networks, such as social and biological interactions, is a central task for understanding and measuring the behavior of these networks. A usual first step in this type of model creation is to construct a digraph with a given degree sequence. We examine the extreme case of digraph construction where for a given degree sequence there is exactly one digraph that can be created.
What follows is a brief introduction to the notation used in the paper. For notation not otherwise defined, see Diestel [1]. We let = ( , ) G V E be a digraph where E is a set of ordered pairs called arcs. If ( , ) v w E ∈ , then we say w is an out-neighbor of v and v is an in-neighbor of w. We notate the out-degree of a vertex v V ∈ by ( ) We are interested in the degree sequences that have unique vertex labeled realizations and the digraphs that realize them. Theorem 1 in Sec. 2 presents several characterizations of this type of degree sequence and its realization. We then show these characterizations to be equivalent. One of the characterizations is previously unpublished, and allows for a much shorter proof of the equivalence of the two known characterizations as well as proving the final characterization which appears without proof in the literature. In Sec. 3, we use Theorem 1 to obtain a new short proof of the Fulkerson-Chen theorem on degree sequences of general digraphs. We end by presenting some applications in Sec. 4

Threshold Digraph Characterization
In the existing literature [2], the characterization of the unique realization of a degree sequence is in terms of forbidden configurations. The two forbidden configurations are the 2-switch and the induced directed 3-cycle. A 2-switch is a set of four vertices , , , w x y z so that ( , ) w x and ( , ) y z are arcs of G and ( , ) w z , ( , ) y x are not. An induced directed 3-cycle is a set of three vertices , , x y z so that ( , ), ( , ), ( , ) x y y z z x are arcs but there are no other arcs among the vertices. Replacement of the arcs in these configurations with the arcs that are not present yields another digraph with the same degrees, both in and out, so any degree sequence of a digraph with these configurations has multiple realizations. These configurations are pictured in Fig. 1.  1. G is the unique labeled realization of the degree sequence α .

2.
There are no 2-switches or induced directed 3-cycles in G . Proof. The equivalence of conditions 1 and 2 has been shown previously [2]. For this proof, we only need to show the implication 1 ⇒ 2, which is shown by the contrapositive: if there were a 2-switch or an induced directed 3-cycle in G , then we can form another graph G′ on the same degree sequence so G does not have a unique realization. Notice that this implication does not require positive lexicographic order.

⇒ 1.
Assume that α is in positive lexicographic order and that we have equality in the Fulkerson-Chen inequalities. We will form the adjacency matrix A one column at a time. Let Since we have equality in the Fulkerson-Chen conditions, we must also have equality for each ( , ) c i k . In We call any digraph that satisfies these conditions threshold. This definition generalizes the wellstudied concept of threshold graphs [3]. http://dx.doi.org/10.6028/jres.119.007 As mentioned above, Rao, Jana, and Bandyopadhyay [2] showed the equivalence of conditions 1 and 2 in the context of Markov chains for generating random zero-one matrices with zero trace. Condition 4 appears in the literature (for example, Berger [4] states this as the definition of threshold digraphs), but we cannot find a proof of its equivalence to the first two conditions. Condition 3 is similar to the criteria of Berger [5,6] stated without proof in the context of corrected Ferrers diagrams.
There are two places where the order of α is important. One is in the statement of condition 4. The second is in the proof of that condition 2 implies condition 3. However, since condition 2 does not depend on the order of the vertices, but on the graph structure, we may characterize threshold digraphs in the absence of the condition that α is in positive lexicographic order. In particular, condition 3 gives that the digraph is threshold even when the degree sequence is unordered.  Thus, A must be the adjacency matrix of G . Since Corollary 3 ties together sequences and threshold digraphs, one application of it is to provide upper and lower bounds on the number of threshold digraphs for a given n . However, if we permute a sequence, then the resulting threshold digraph may or may not be isomorphic. For example, on three vertices the six orders of the sequence (2,1, 0) produce two non-isomorphic threshold digraphs. The sequences (2,1, 0), (1, 2, 0) , and (2, 0,1) all produce the same digraph with degree sequence ((1, 2), (1,1), (1, 0)) in positive lexicographic order, while the remaining three sequences produce the threshold digraph with degree sequence ((2, 0), (1,1), (0, 2)) in positive lexicographic order.

Digraph Realizability
The idea of condition 4 comes from what are known as the Fulkerson-Chen inequalities for digraph realizability. Fulkerson studied digraph realizability in the context of zero-one matrices with zero trace [7]. For a given degree sequence, Fulkerson gave a system of 2 1 n − inequalities that are satisfied if and only if the degree sequence is digraphical. The formulation that we typically use is due to Chen [8], which reduces the number of inequalities from 2 1 n − to n when the degree sequence is in negative lexicographic order. Our consideration of threshold digraphs gives a new proof of this result.
This proof uses the partial order ≼, commonly called majorization [9], on integer sequences. In as desired.
Suppose that α is a sequence which satisfies the above inequalities. Construct an adjacency matrix T as in Corollary 3 from the sequence α − . We will iteratively form a sequence of digraphs otherwise.
Clearly ( ) ≻ ( +1) ≽ + . Since Therefore, is a realization of α , as desired. This proof is constructive; given a digraphical degree sequence α , we can construct a realization of α by repeatedly moving the ones down in the columns as in the proof of Theorem 5. There are other construction algorithms for digraphs, most notably that of Kleitman and Wang [10].

Applications
What follows is a quick survey of some consequences of Theorem 1. Some details are omitted since the first two results are immediate.
Threshold graphs, in the undirected sense, are closely tied to the theory of split graphs. An analogous study of split digraphs is given by LaMar [11]. Using the fourth characterization of threshold digraphs and a result by LaMar, the immediate implication is Corollary 6.

Corollary 6. Every threshold digraph is a split digraph.
Merris and Roby [12] studied the relationship between different threshold graphs as subgraphs of one another. As a simple consequence of the third characterization of threshold digraphs, there is a similar relationship between threshold digraphs which we state as Corollary 7.

Corollary 7. Given a threshold digraph G , if G is nonempty, then there is an arc e in G such that G e
− is a threshold digraph. If G is not complete, then there is an arc e not in G such that G e + is a threshold digraph.
It has been observed that the ordering required by Theorem 5 can be relaxed and still only require the n inequalities stated. Berger [4] observed that we need only require nonincreasing order in the first component. Our theorem suggests that this can be relaxed even more, but it is not readily apparent which orders should be considered for graphicality. However, we can show a simple proof that nonincreasing order in the first component is sufficient. This section gives a brief overview of some of the applications of threshold digraphs. The uses of threshold graphs in various disciplines has been studied extensively, as shown in Mahadev and Peled's text [3]. These results are a starting point for an analogous study of threshold digraphs.