A Characterization of Interval Catch Digraphs

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, 1988

Given an interval representation of an interval graph G, an interval is an end interval if its right (or left) endpoint is further to the right (left) of all other intervals in the collection. We characterize those vertices v in interval graphs for which there is some representation of G where the interval corresponding to v is an end interval. We also present a short proof of a characterization of homogeneously representable interval graphs. <

ArXiv, 2021

Given a digraph G, a set X ⊆ V (G) is said to be an absorbing set (resp. dominating set) if every vertex in the graph is either in X or is an in-neighbour (resp. out-neighbour) of a vertex in X. A set S ⊆ V (G) is said to be an independent set if no two vertices in S are adjacent in G. A kernel (resp. solution) of G is an independent and absorbing (resp. dominating) set in G. The problem of deciding if there is a kernel (or solution) in an input digraph is known to be NP-complete. Similarly, the problems of computing a minimum cardinality kernel, absorbing set (or dominating set) and the problems of computing a maximum cardinality kernel, independent set are all known to be NP-hard for general digraphs. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph G is an interval digraph if a pair of intervals (Su, Tu) can be assigned to each vertex u of G such that (u, v) ∈ E(G) if and only if Su ∩ Tv 6= ∅. Many different subclasse...

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.

Discrete Applied Mathematics, 1998

Probe interval graphs have been introduced in the physical mapping and sequencing of DNA as a generalization of interval graphs. We prove that probe interval graphs are weakly triangulated, and hence are perfect, and characterize probe interval graphs by consecutive orders of their intrinsic cliques.

SIAM Journal on Discrete Mathematics, 2012

We introduce a class of digraphs analogous to proper interval graphs and bigraphs. They are defined via a geometric representation by two inclusion-free families of intervals satisfying a certain monotonicity condition; hence we call them monotone proper interval digraphs. They admit a number of equivalent definitions, including an ordering characterization by so-called Min-Max orderings, and the existence of certain graph polymorphisms. Min-Max orderings arose in the study of minimum cost homomorphism problems: if H admits a a Min-Max ordering (or a certain extension of Min-Max orderings), then the minimum cost homomorphism problem to H is known to admit a polynomial time algorithm. We give a forbidden structure characterization of monotone proper interval digraphs, which implies a polynomial time recognition algorithm. This characterizes digraphs with a Min-Max ordering; we also similarly characterize digraphs with an extended Min-Max ordering. In a companion paper, we shall apply this latter characterization to derive a conjectured dichotomy classification for the minimum cost homomorphism problems-namely, we shall prove that the minimum cost homomorphism problem to a digraph that does not admit an extended Min-Max ordering is NP-complete.

2017

An interval containment model of a graph maps vertices into intervals of a line in such a way that two vertices are adjacent if and only if the corresponding intervals are comparable under the inclusion relation. Graphs admitting an interval containment model are called containment interval graphs or CI graphs for short. A vertex v of a CI graph G is an end-vertex if there is an interval containment model of G in which the left endpoint of the interval corresponding to v is less than all other endpoints. In this work, we present a characterization of end-vertices in terms of forbidden induced subgraphs.

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.

Combinatorica, 2014

We resolve two problems of [Cameron, Praeger, and Wormald -Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each 'building block' is a finite bipartite graph that is not a disjoint union of complete bipartite graphs. This was conjectured impossible in the above paper. We also describe the structure of 2-ended highly arc transitive digraphs in more generality, although complete characterization remains elusive.

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.