Intersection Digraph Research Articles

Classes of intersection digraphs with good algorithmic properties

AbstractWhile intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection digraphs in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied in the literature. Second, we define the directed locally checkable vertex (DLCV) problems, which capture many well‐studied problems on digraphs, such as (Independent) Dominating Set, Kernel, and ‐Homomorphism. Third, we give a new width measure of digraphs, bi‐mim‐width, and show that the DLCV problems are polynomial‐time solvable when we are provided a decomposition of small bi‐mim‐width. Fourth, we show that several classes of intersection digraphs have bounded bi‐mim‐width, implying that we can solve all DLCV problems on these classes in polynomial time given an intersection representation of the input digraph. We identify reflexivity as a useful condition to obtain intersection digraph classes of bounded bi‐mim‐width, and therefore to obtain positive algorithmic results.

On small subgraphs in a random intersection digraph

Given a set of vertices V and a set of attributes W let each vertex v∈V include an attribute w∈W into a set S−(v) with probability p− and let it include w into a set S+(v) with probability p+ independently for each w∈W. The random binomial intersection digraph on the vertex set V is defined as follows: for each u,v∈V the arc uv is present if S−(u) and S+(v) are not disjoint. For any h=2,3,… we determine the birth threshold of the complete digraph on h vertices and describe the configurations of intersecting sets that realise the threshold.

The distribution of the relative arc density of a family of interval catch digraph based on uniform data

We study a family of interval catch digraph called proportional-edge proximity catch digraph (PCD) which is also a special type of intersection digraphs parameterized with an expansion and a centrality parameter. PCDs are random catch digraphs that have been developed recently and have applications in classification and spatial pattern analysis. We investigate a graph invariant of the PCDs called relative arc density. We demonstrate that relative arc density of PCDs is a U-statistic and using the central limit theory of U-statistics, we derive the (asymptotic) distribution of the relative arc density of proportional-edge PCD for uniform data in one dimension. We also determine the parameters for which the rate of convergence to asymptotic normality is fastest.

A random intersection digraph: Indegree and outdegree distributions

Let S ( 1 ) , … , S ( n ) , T ( 1 ) , … , T ( n ) be random subsets of the set [ m ] = { 1 , … , m } . We consider the random digraph D on the vertex set [ n ] defined as follows: the arc i → j is present in D whenever S ( i ) ∩ T ( j ) ≠ 0̸ . Assuming that the pairs of sets ( S ( i ) , T ( i ) ) , 1 ≤ i ≤ n , are independent and identically distributed, we study the in- and outdegree distributions of a typical vertex of D as n , m → ∞ .

Chronological orderings of interval digraphs

An interval digraph is the intersection digraph of a family of ordered pairs of intervals on the real line; the family is an interval representation of the digraph. It is well known that an interval digraph has many representations that differ in the order of the endpoints of the intervals on the line. This paper generalizes the corresponding results on interval graphs by Skrien [Chronological orderings of interval graphs, Discrete Appl. Math. 8 (1984) 69–83] and describes how, given an interval digraph, the order of intervals of one representation differs from another.

On the distribution of the domination number for random class cover catch digraphs

In this article we initiate the study of class cover catch digraphs, a special case of intersection digraphs motivated by applications in machine learning and statistical pattern recognition. Our main result is the exact distribution of the domination number for a data-driven model of random interval catch digraphs.

Line digraphs and coreflexive vertex sets

The concept of coreflexive set is introduced to study the structure of digraphs. New characterizations of line digraphs and nth-order line digraphs are given. Coreflexive sets also lead to another natural way of forming an intersection digraph from a given digraph.

Subpath acyclic digraphs

A characterization is given for acyclic digraphs that are the acyclic intersection digraphs of subpaths of a directed tree.

Representing digraphs using intervals or circular arcs

Containment and overlap representations of digraphs are studied, with the following results. The interval containment digraphs are the digraphs of Ferrers dimension 2, and the circular-arc containment digraphs are the complements of circular-arc intersection digraphs. A poset is an interval containment poset if and only if its comparability digraph is an interval containment digraph, and a graph is an interval graph if and only if the corresponding symmetric digraph with loops is an interval digraph. In an appropriate model of overlap representation using intervals, the unit right-overlap interval digraphs are precisely the unit interval digraphs, and the adjacency matrices of right-overlap interval digraphs have a simple structural characterization bounding their Ferrers dimension by 3.

A note on recursive properties of the de Bruijn, Kautz and FFT digraphs

Abstract It is well known that the de Bruijn digraph (and also the Kautz digraph) of in- and out-degree d and diameter n is the line-digraph of the same digraph of diameter n − 1. We show here that the Bruijn digraph of diameter n is the intersection digraph of complete bipartite digraphs of the de Bruijn digraph of diameter n + 1. The same property occurs for Kautz and Fast-Fourier-Transform digraphs.

An interval digraph in relation to its associated bipartite graph

The intersection digraph of a family of ordered pairs of sets {( S v , T v ): v∈ V} is the digraph D( V, E) such that uv∈ E if and only if S u ∩ T v ≠0. Internal digraphs are those intersection digraphs for which the subsets are intervals on the real line. In a previous paper, they were characterized in terms of Ferrers digraphs and a close relationship was obtained between an interval digraph and a digraph of Ferrers dimension 2. In order to characterize a digraph D of Ferrers dimension 2, Cogis associated an undirected graph H( D) with D in a suitable way, the vertices of H( D) corresponding to the zeros of the adjacency matrix of D. He proved that D has Ferrers dimension at most 2 if and only if H( D) is bipartite. Depending on the above characterization, this paper first obtains some properties of a digraph of Ferrers dimension 2; then it is shown how the notion of interior edges is related to an interval digraph.

Circular‐arc digraphs: A characterization

AbstractGiven a family F of ordered pairs of sets {(Su,Tu): u ϵ V}, the intersection digraph of F is the digraph with vertices V defined by uv ϵ E(D) if Su ≠ Tv Ø. Interval digraphs are intersection digraphs of families in which every set is an interval, and were characterized in a previous paper by conditions on the adjacency matrix. In this paper, another such condition leads to an adjacency matrix characterization of circular‐arc digraphs, the intersection digraphs of families in which every set is an arc of a circle.

Interval digraphs: An analogue of interval graphs

AbstractIntersection digraphs analogous to undirected intersection graphs are introduced. Each vertex is assigned an ordered pair of sets, with a directed edge uv in the intersection digraph when the “source set” of u intersects the “terminal set” of v. Every n‐vertex digraph is an intersection digraph of ordered pairs of subsets of an n‐set, but not every digraph is an intersection digraph of convex sets in the plane. Interval digraphs are those having representations where all sets are intervals on the real line. Interval digraphs are characterized in terms of the consecutive ones property of certain matrices, in terms of the adjacency matrix and in terms of Ferrers digraphs. In particular, they are intersections of pairs of Ferrers digraphs whose union is a complete digraph.