End vertices in containment interval graphs

arXiv (Cornell University), 2018

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, threshold graphs, complements of threshold tolerance graphs (known as 'co-TT' graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray graphs. (The last three classes coincide, but have been investigated in different contexts.) This common view is made possible by introducing reflexive relationships (loops) into the analysis. We also show that all the above classes are united by a common ordering characterization, the existence of a min ordering. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, and show that they are precisely the digraphs that are characterized by the existence of a min ordering. We also offer an alternative geometric characterization of these digraphs. For most of the above graph and digraph classes, we show that they are exactly those signed-interval digraphs that satisfy a suitable natural restriction on the digraph, like having a loop on every vertex, or having a symmetric edge-set, or being bipartite. For instance, co-TT graphs are precisely those signed-interval digraphs that have each edge symmetric. We also offer some discussion of future work on recognition algorithms and characterizations.

Bull. Math. Soc. Sci. Math. Roumanie, 2024

A graph G is called an interval if there exist a, b ∈ V (G) such that G is the union of shortest paths connecting a and b. In this paper we show that 1. If G is an interval between a and b, then there exists a path H with diameter d(H) = d(G) such that there is a homomorphism f : G → H and the distance ρ(a, b) + 1 ≤ |H| ≤ |G|-1; 2. Every interval is a connected bipartite graph; 3. If G is an interval between a and b that is not a path, then G has a path P with internal vertices (if any) all of degree 2 such that deletion of the internal vertices of P from G gives rise to an interval (if P = uv then G-uv is an interval).

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.

Journal of graph theory, 1989

Intersection digraphs analogous to undirected intersection graphs are introduced. Each vertex is assigned an ordered pair of sets, with a directed edge uu in the intersection digraph when the "source set" of u intersects the "terminal set" of u. 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 o f ' Ferrers digraphs whose union is a complete digraph.

2020

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V,E) with vertex set V and edge set E, a labeling φ : V ∪E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x ∈ V (G) under a total k-labeling φ is defined as wt(x) = φ(x)+ ∑ y∈N(x) φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph G, if wt(x) 6= wt(y) holds for every two different vertices x and y of G. The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs are presented for interval graphs.

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.

Journal of Scientific Research, 2020

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V, E) with vertex set V and edge set E, a labeling Φ : V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling Φ is defined as () () ∑ () ∈ () where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

Journal of Graph Theory, 2004

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, denoted by i(G). Griggs and West showed that i(G) ≤ 1 2 (d + 1) . We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independence number. Finally we show that a graph needs to contain large complete bipartite induced subgraphs in order to have interval number larger than the random graph on the same number of vertices.

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 .

WG'07, 2007

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatics problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K 1,5-free graphs, ...