Interval digraphs: an analogue of interval graphs

The Electronic Journal of Combinatorics, 2013

Interval graphs admit elegant structural characterizations and linear time recognition algorithms; on the other hand, the usual interval digraphs lack a forbidden structure characterization as well as a low-degree polynomial time recognition algorithm. In this paper we identify another natural digraph analogue of interval graphs that we call ”chronological interval digraphs”. By contrast, the new class admits both a forbidden structure characterization and a linear time recognition algorithm. Chronological interval digraphs arise by interpreting the standard definition of an interval graph with a natural orientation of its edges. Specifically, $G$ is a chronological interval digraph if there exists a family of closed intervals $I_v$, $v \in V(G)$, such that $uv$ is an arc of $G$ if and only if $I_u$ intersects $I_v$ and the left endpoint of $I_u$ is not greater than the left endpoint of $I_v$. (Equivalently, if and only if $I_u$ contains the left endpoint of $I_v$.)We characterize c...

Discrete Applied Mathematics, 2012

Interval graphs admit linear time recognition algorithms and have several elegant forbidden structure characterizations. Interval digraphs can also be recognized in polynomial time and admit a characterization in terms of incidence matrices. Nevertheless, they do not have a known forbidden structure characterization or low-degree polynomial time recognition algorithm.

arXiv (Cornell University), 2011

An interval graph is proper iff it has a representation in which no interval contains another. Fred Roberts [27] characterized the proper interval graphs as those containing no induced star K1,3. Proskurowski and Telle [26] have studied q-proper graphs, which are interval graphs having a representation in which no interval is properly contained in more than q other intervals. Like Roberts they found that their classes of graphs where characterized, each by a single minimal forbidden subgraph. This paper initiates the study of p-improper interval graphs where no interval contains more than p other intervals. This paper will focus on a special case of p-improper interval graphs for which the minimal forbidden subgraphs are readily described. Even in this case, it is apparent that a very wide variety of minimal forbidden subgraphs are possible.

Journal of Graph Theory, 1979

In this paper w e discuss a generalization of the familiar concept of an interval graph that arises naturally in scheduling and allocation problems. We define the interval number of a graph G to be the smallest positive integer t for which there exists a function f which assigns to each vertex u of G a subset f(u) of the real line so that f(u) is the union of t closed intervals of the real line, and distinct vertices u and v in G are adjacent if and only if f(u) and f (v) meet. We show that (1) the interval number of a tree is a t most two, and (2) the complete bipartite graph Km,n has interval number [(mn + 1) / (m + n) l .

Algorithmica

In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called k-length interval graphs were considered in which the number of different lengths of intervals is limited by k. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for k = 2. We propose generalizations of proper interval graphs called k-nested interval graphs in which there are no chains of k + 1 intervals nested in each other. It is easy to see that k-nested interval graphs are a superclass of k-length interval graphs. We give a linear-time recognition algorithm for k-nested interval graphs. This algorithm adds a missing piece to Gajarský et al. [FOCS 2015] to show that testing FO properties on interval graphs is FPT with respect to the nesting k and the length of the formula, while the problem is W[2]-hard when parameterized just by the length of the formula. Further, we show that a generalization of recognition called partial representation extension is polynomial-time solvable for k-nested interval graphs, while it is NP-hard for k-length interval graphs, even when k = 2.

SIAM Journal on Discrete Mathematics, 2020

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, complements of threshold tolerance graphs (known as co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. (The last three classes coincide, but have been investigated in different contexts.) We show that all of the above classes are united by a common ordering characterization, the existence of a min ordering. However, because the presence or absence of reflexive relationships (loops) affect whether a graph or digraph has a min ordering, to obtain this result, we must define the graphs and digraphs to have those loops that are implied by their definitions. These have been largely ignored in previous work. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, characterized by the existence of a compact representation, a signed-interval model, which is a generalization of known representations of the graph classes. We show that the signed-interval digraphs are precisely those digraphs that are characterized by the existence of a min ordering when the loops implied by the model are considered part of the graph. We also offer an alternative geometric characterization of these digraphs. We show that co-TT graphs are the symmetric signed-interval digraphs, the adjusted interval digraphs are the reflexive signed-interval digraphs, and the interval graphs are the intersection of these two classes, namely, the reflexive and symmetric signed-interval digraphs. Similar results hold for bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. 1

2021

We introduce a novel framework of graph modifications specific to interval graphs. We study interdiction problems with respect to these graph modifications. Given a list of original intervals, each interval has a replacement interval such that either the replacement contains the original, or the original contains the replacement. The interdictor is allowed to replace up to k original intervals with their replacements. Using this framework we also study the contrary of interdiction problems which we call assistance problems. We study these problems for the independence number, the clique number, shortest paths, and the scattering number. We obtain polynomial time algorithms for most of the studied problems. Via easy reductions, it follows that on interval graphs, the most vital nodes problem with respect to shortest path, independence number and Hamiltonicity can be solved in polynomial time.

Lecture Notes in Computer Science, 2014

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem.

Acta Cybernetica

The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here, we propose a family of representations for the graph G, which yield the well-known upper bound ⌈1 2(d+1)⌉, where d is the maximum degree of G. The extremal graphs for even d are also described, and the upper bound on the interval number in terms of the number of edges of G is improved.

Discrete Mathematics, 1989