Integral mixed unit interval graphs

We characterize graphs that have intersection representations using unit intervals with open or closed ends such that all ends of the intervals are integral in terms of infinitely many minimal forbidden induced subgraphs. Furthermore, we provide a linear-time algorithm that decides if a given interval graph admits such an intersection representation.     

Completion of the mixed unit interval graphs hierarchy

We describe the missing class of the hierarchy of mixed unit interval graphs. This class is generated by the intersection graphs of families of unit intervals that are allowed to be closed, open, and left‐closed‐right‐open. (By symmetry, considering closed, open, and right‐closed‐left‐open unit intervals generates the same class.) We show that this class lies strictly between unit interval graphs and mixed unit interval graphs. We give a complete characterization of this new class, as well as quadratic‐time algorithms that recognize graphs from this class and produce a corresponding interval representation if one exists. We also show that the algorithm from Shuchat et al. [8] directly extends to provide a quadratic‐time algorithm to recognize the class of mixed unit interval graphs.     

A Characterization of Mixed Unit Interval Graphs

We give a complete characterization of mixed unit interval graphs, the intersection graphs of closed, open, and half‐open unit intervals of the real line. This is a proper superclass of the well‐known unit interval graphs. Our result solves a problem posed by Dourado, Le, Protti, Rautenbach, and Szwarcfiter (Mixed unit interval graphs, Discrete Math 312, 3357–3363 (2012)).     

Unit Interval Graphs of Open and Closed Intervals

We give two structural characterizations of the class of finite intersection graphs of the open and closed real intervals of unit length. This class is a proper superclass of the well‐known unit interval graphs.     

Interval <Emphasis Type="Italic">k</Emphasis>-Graphs and Orders

An interval k-graph is the intersection graph of a family of intervals of the real line partitioned into k classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we study the cocomparability interval k-graphs; that is, the interval k-graphs whose complements have a transitive orientation and are therefore the incomparability graphs of strict partial orders. For brevity we call these orders interval k-orders. We characterize the kind of interval representations a cocomparability interval k-graph must have, and identify the structure that guarantees an order is an interval k-order. The case k =?2 is peculiar: cocomparability interval 2-graphs (equivalently proper- or unit-interval bigraphs, bipartite permutation graphs, and complements of proper circular-arc graphs to name a few) have been characterized in many ways, but we show that analogous characterizations do not hold if k >?2. We characterize the cocomparability interval 3-graphs via one forbidden subgraph and hence interval 3-orders via one forbidden suborder.     

Partial Characterizations of Circular-Arc Graphs

A circular-arc graph is the intersection graph of a family of arcs on a circle. A characterization by forbidden induced subgraphs for this class of graphs is not known, and in this work we present a partial result in this direction. We characterize circular-arc graphs by a list of minimal forbidden induced subgraphs when the graph belongs to the following classes: diamond-free graphs, P₄-free graphs, paw-free graphs, and claw-free chordal graphs.     

Diamond-free circle graphs are Helly circle

The diamond is the graph obtained from K₄ by deleting an edge. Circle graphs are the intersection graphs of chords in a circle. Such a circle model has the Helly property if every three pairwise intersecting chords intersect in a single point, and a graph is Helly circle if it has a circle model with the Helly property. We show that the Helly circle graphs are the diamond-free circle graphs, as conjectured by Durán. This characterization gives an efficient recognition algorithm for Helly circle graphs.     

Coloring intersection graphs of x-monotone curves in the plane

A class of graphs G is χ-bounded if the chromatic number of the graphs in G is bounded by some function of their clique number. We show that the class of intersection graphs of simple families of x-monotone curves in the plane intersecting a vertical line is χ-bounded. As a corollary, we show that the class of intersection graphs of rays in the plane is χ-bounded, and the class of intersection graphs of unit segments in the plane is χ-bounded.     

Asteroids in rooted and directed path graphs

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it contains no asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a strong path. Two non-adjacent vertices are linked by a strong path if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain conditions. A strong asteroidal triple is an asteroidal triple such that each pair is linked by a strong path. We prove that a chordal graph is a directed path graph if and only if it contains no strong asteroidal triple. We also introduce a related notion of asteroidal quadruple, and conjecture a characterization of rooted path graphs which are the intersection graphs of directed paths in a rooted tree.     

Relationships between the class of unit grid intersection graphs and other classes of bipartite graphs

We show that the class of unit grid intersection graphs properly includes both of the classes of interval bigraphs and of P₆-free chordal bipartite graphs. We also demonstrate that the classes of unit grid intersection graphs and of chordal bipartite graphs are incomparable.     

Tree loop graphs

Many problems involving DNA can be modeled by families of intervals. However, traditional interval graphs do not take into account the repeat structure of a DNA molecule. In the simplest case, one repeat with two copies, the underlying line can be seen as folded into a loop. We propose a new definition that respects repeats and define loop graphs as the intersection graphs of arcs of a loop. The class of loop graphs contains the class of interval graphs and the class of circular-arc graphs. Every loop graph has interval number 2. We characterize the trees that are loop graphs. The characterization yields a polynomial-time algorithm which given a tree decides whether it is a loop graph and, in the affirmative case, produces a loop representation for the tree.     

Tolerance graphs

Tolerance graphs arise from the intersection of intervals with varying tolerances in a way that generalizes both interval graphs and permutation graphs. In this paper we prove that every tolerance graph is perfect by demonstrating that its complement is perfectly orderable. We show that a tolerance graph cannot contain a chordless cycle of length greater than or equal to 5 nor the complement of one. We also discuss the subclasses of bounded tolerance graphs, proper tolerance graphs, and unit tolerance graphs and present several possible applications and open questions.     

Unit and single point interval graphs

We describe a linear time algorithm for the recognition of graphs that have an intersection representation using unit length intervals and single point intervals. Furthermore, we characterize these graphs using forbidden induced subgraphs.     

A characterization of cycle-free unit probe interval graphs

A graph is a probe interval graph (PIG) if its vertices can be partitioned into probes and nonprobes with an interval assigned to each vertex so that vertices are adjacent if and only if their corresponding intervals overlap and at least one of them is a probe. PIGs are a generalization of interval graphs introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. PIGs have been characterized in the cycle-free case by Sheng, and other miscellaneous results are given by McMorris, Wang, and Zhang. Johnson and Spinrad give a polynomial time recognition algorithm for when the partition of vertices into probes and nonprobes is given. The complexity for the general recognition problem is not known. Here, we restrict attention to the case where all intervals have the same length, that is, we study the unit probe interval graphs and characterize the cycle-free graphs that are unit probe interval graphs via a list of forbidden induced subgraphs.     

A proof of Hougardy's conjecture for diamond-free graphs

A pair of vertices of a graph is called an even pair if every chordless path between them has an even number of edges. A graph is minimally even pair free if it is not a clique, contains no even pair, but every proper induced subgraph either contains an even pair or is a clique. Hougardy (European J. Combin. 16 (1995) 17–21) conjectured that a minimally even pair free graph is either an odd cycle of length at least five, the complement of an even or odd cycle of length at least five, or the linegraph of a bipartite graph. A diamond is a graph obtained from a complete graph on four vertices by removing an edge. In this paper we verify Hougardy's conjecture for diamond-free graphs by adapting the characterization of perfect diamond-free graphs given by Fonlupt and Zemirline (Maghreb Math. Rev. 1 (1992) 167–202).     

Graph limits and hereditary properties

We give a survey of some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.     

Random interval graphs

In this paper we introduce a notion ofrandom interval graphs: the intersection graphs of real, compact intervals whose end points are chosen at random. We establish results about the number of edges, degrees, Hamiltonicity, chromatic number and independence number of almost all interval graphs.     

Characterizing directed path graphs by forbidden asteroids

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it does not contain an asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a special connection. Two non‐adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex‐disjoint chordless paths satisfying certain conditions. A special asteroidal triple is an asteroidal triple such that each pair is linked by a special connection. We prove that a chordal graph is a directed path graph if and only if it does not contain a special asteroidal triple. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:103‐112, 2011     

Edge and vertex intersection of paths in a tree

In this paper we continue the investigation of the class of edge intersection graphs of a collection of paths in a tree (EPT graphs) where two paths edge intersect if they share an edge. The class of EPT graphs differs from the class known as path graphs, the latter being the class of vertex intersection graphs of paths in a tree. A characterization is presented here showing when a path graph is an EPT graph. In particular, the classes of path graphs and EPT graphs coincide on trees all of whose vertices have degree at most 3. We then prove that it is an NP-complete problem to recognize whether a graph is an EPT graph.     

Algorithmic aspects of intersection graphs and representation hypergraphs

Let be a family of sets. The intersection graph of is obtained by representing each set in by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersect. Of primary interest are those classes of intersection graphs of families of sets having some specific topological or other structure. The grandfather of all intersection graphs is the class of interval graphs, that is, the intersection graphs of intervals on a line.The scope of research that has been going on in this general area extends from the mathematical and algorithmic properties of intersection graphs, to their generalizations and graph parameters motivated by them. In addition, many real-world applications involve the solution of problems on such graphs.In this paper a number of topics in algorithmic combinatorics which involve intersection graphs and their representative families of sets are presented. Recent applications to computer science are also discussed. The intention of this presentation is to provide an understanding of the main research directions which have been investigated and to suggest possible new directions of research.