Class Of Intersection Graphs Research Articles

A characterization of unit interval bigraphs of open and closed intervals

Interval graphs are the intersection graphs of intervals on the real line. Unit interval graphs are interval graphs representable by intervals of unit length. Rautenbach and Szwarcfiter showed that the class of intersection graphs of the open and closed real intervals of unit length is a strict superclass of the class of unit interval graphs. They also characterized this class of graphs.An interval bigraph is a bipartite graph where to each vertex we can assign an interval so that vertices in the opposite partite sets are adjacent if and only if their intervals intersect. Analogously we show that the class of finite intersection bigraphs of open and closed intervals of unit length is a strict superclass of the class of unit interval bigraphs. We also characterize this class of bigraphs in terms of forbidden induced subgraphs and a suitable extension of proper interval bigraphs.

Finding geometric representations of apex graphs is NP-hard

Planar graphs can be represented as the intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., SODA 2018). For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex graphs, i.e., graphs that can be made planar by removing one vertex from them. We show, somewhat surprisingly, that deciding whether geometric representations exist for apex graphs is NP-hard as well.More precisely, we show that for every fixed positive integer g and every graph class G such that ▪, it is NP-hard to decide whether an input graph belongs to the graph class G, even when the inputs are restricted to apex graphs of girth g. Here, ▪ is the class of intersection graphs of axis-parallel line segments (where horizontal segments intersect only vertical segments), and ▪ is the class of intersection graphs of simple curves (where two intersecting curves cross each other exactly once) in the plane. This partially answers an open question raised by Kratochvíl & Pergel (COCOON, 2007).Most known reductions for earlier proofs of NP-hardness for these problems are from variants of 3-SAT (mainly Planar 3-Connected 3-SAT). We reduce from the ▪▪▪ problem, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric intersection graphs.

How heavy independent sets help to find arborescences with many leaves in DAGs

Trees with many leaves have applications for broadcasting a message to all recipients simultaneously. Internal nodes of a broadcasting tree require more expensive technology to forward the messages received. We address a problem that captures the main goal: finding spanning trees with few internal nodes in a given network. The Maximum Leaf Spanning Arborescence problem consists of, given a directed graph D, finding a spanning arborescence of D, if one exists, with the maximum number of leaves. This problem is NP-hard in general and MaxSNP-hard in the rooted directed acyclic graphs class. This paper explores a relationship between Maximum Leaf Spanning Arborescence in rooted directed acyclic graphs and maximum weight set packing. The latter problem is related to independent sets on particular classes of intersection graphs. Exploiting this relation, we derive a 7/5-approximation for Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs, improving on the previous 3/2-approximation.

What Graphs are 2-Dot Product Graphs?

Let [Formula: see text] be an integer. From a set of [Formula: see text]-dimensional vectors, we obtain a [Formula: see text]-dot product graph by letting each vector [Formula: see text] correspond to a vertex [Formula: see text] and by adding an edge between two vertices [Formula: see text] and [Formula: see text] if and only if their dot product [Formula: see text], for some fixed, positive threshold [Formula: see text]. Dot product graphs can be used to model social networks. Recognizing a [Formula: see text]-dot product graph is known to be [Formula: see text]-hard for all fixed [Formula: see text]. To understand the position of [Formula: see text]-dot product graphs in the landscape of graph classes, we consider the case [Formula: see text], and investigate how [Formula: see text]-dot product graphs relate to a number of other known graph classes including a number of well-known classes of intersection graphs.

On the speed of algebraically defined graph classes

The speed of a class of graphs counts the number of graphs on the vertex set {1,…,n} inside the class as a function of n. In this paper, we investigate this function for many classes of graphs that naturally arise in discrete geometry, for example intersection graphs of segments or disks in the plane. While upper bounds follow from Warren's theorem (a variant of a theorem of Milnor and Thom), all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving an essentially tight lower bound for the number of graphs on {1,…,n} whose edges are defined using the signs of a given finite list of polynomials, assuming these polynomials satisfy some reasonable conditions. This in particular implies lower bounds for the speed of many different classes of intersection graphs, which essentially match the known upper bounds. Our general result also gives essentially tight lower bounds for counting containment orders of various families of geometric objects, including circle orders and angle orders. Some of the applications presented in this paper are new, whereas others recover results of Alon-Scheinerman, Fox, McDiarmid-Müller and Shi. For the proof of our result we use some tools from algebraic geometry and differential topology.

Conflict-Free Coloring of Intersection Graphs of Geometric Objects

In 2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflict-free colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with $$O(\log n)$$ colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we obtain the following strengthening of the two main results of Even et al.: Any family $$\mathcal {F}$$ of n discs in the plane can be colored with $$O(\log n)$$ colors in such a way that for any disc B in the plane, not necessarily from $$\mathcal {F}$$ , the set of discs in $$\mathcal {F}$$ that intersect B contains a uniquely-colored element. In view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of $$\rho $$ -fat objects in the plane.

Polygon-circle and word-representable graphs

We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs.A graph G = (V, E) is word-representable if there exists a word w over the alpha-bet V such that letters x and y form a subword of the form xyxy ⋯ or yxyx ⋯ iff xy is an edge in E. Word-representable graphs generalise several well-known and well-studied classes of graphs [S. Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, Lecture Notes in Computer Science 10396 (2017) 36–67; S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015]. It is known that any word-representable graph is k-word-representable, that is, can be represented by a word having exactly k copies of each letter for some k dependent on the graph. Recognising whether a graph is word-representable is NP-complete ([S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015, Theorem 4.2.15]). A polygon-circle graph (also known as a spider graph) is the intersection graph of a set of polygons inscribed in a circle [M. Koebe, On a new class of intersection graphs, Ann. Discrete Math. (1992) 141–143]. That is, two vertices of a graph are adjacent if their respective polygons have a non-empty intersection, and the set of polygons that correspond to vertices in this way are said to represent the graph. Recognising whether an input graph is a polygon-circle graph is NP-complete [M. Pergel, Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete, Graph-Theoretic Concepts in Computer Science: 33rd Int. Workshop, Lecture Notes in Computer Science, 4769 (2007) 238–247]. We show that neither of these two classes is included in the other one by showing that the word-representable Petersen graph and crown graphs are not polygon-circle, while the non-word-representable wheel graph W5 is polygon-circle. We also provide a more refined result showing that for any k ≥ 3, there are k-word-representable graphs which are neither (k −1)-word-representable nor polygon-circle.

The complexity of Dominating Set in geometric intersection graphs

We study the parameterized complexity of the dominating set problem in geometric intersection graphs.•In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size).•In two and higher dimensions, we prove that Dominating Set is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. So far this was only known for unit squares. Finally, we establish W[1]-hardness for a large class of intersection graphs.

Coloring Curves that Cross a Fixed Curve

We prove that for every integer tgeqslant 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is chi -bounded. This is essentially the strongest chi -boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers kgeqslant 2 and tgeqslant 1, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has O(nlog n) edges.

Cops and Robbers on intersection graphs

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: •The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g+15 in case of orientable surface of genus g, and at most 10g′+15 in case of non-orientable surface of Euler genus g′. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4.•The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, it follows from known results that the cop number is unbounded on intersection graphs of two-element subsets of a line. We further show that it is also unbounded on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

Homothetic polygons and beyond: Maximal cliques in intersection graphs

We study the structure and the maximum number of maximal cliques in classes of intersection graphs of convex sets in the plane. It is known that convex-set intersection graphs, and also straight-line-segment intersection graphs may have exponentially many maximal cliques. On the other hand, in intersection graphs of homothetic triangles, the maximum number of maximal cliques is polynomial in the number of vertices. We extend the latter result by showing that for every convex polygon P with sides parallel to k directions, every n-vertex graph which is an intersection graph of homothetic copies of P contains at most nk inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so-called kDIR-CONV, which are intersection graphs of convex polygons whose sides are parallel to some fixed k directions. Moreover, we provide lower bounds on the maximum number of maximal cliques and generalize the upper bound to intersection graphs of higher-dimensional convex polytopes in Euclidean space. Finally, we discuss the algorithmic consequences of the polynomial bound on the number of maximal cliques.

Intersection Graphs of Rays and Grounded Segments

We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class, not every intersection graph of rays is an intersection graph of downward rays, and not every outer segment graph is an intersection graph of rays. The first result answers an open problem posed by Cabello and Jejčič. The third result confirms a conjecture by Cabello. We thereby completely elucidate the remaining open questions on the containment relations between these classes of segment graphs. We further characterize the complexity of the recognition problems for the classes of outer segment, grounded segment, and ray intersection graphs. We prove that these recognition problems are complete for the existential theory of the reals. This holds even if a 1-string realization is given as additional input.

Refining the Hierarchies of Classes of Geometric Intersection Graphs

We analyse properties of geometric intersection graphs to show the strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties:A graph $G$ is outerplanar if and only if the 1-subdivision of $G$ is outer-segment.For each integer $k\ge 1$, the class of intersection graphs of segments with $k$ different lengths is a strict subclass of the class of intersection graphs of segments with $k+1$ different lengths.For each integer $k\ge 1$, the class of intersection graphs of disks with $k$ different sizes is a strict subclass of the class of intersection graphs of disks with $k+1$ different sizes.The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

A Note on Fractional Coloring and the Integrality gap of LP for Maximum Weight Independent Set

We prove a tight connection between two important notions in combinatorial optimization. Let G be a graph class (i.e. a subset of all graphs) and r(G)=supG∈G⁡χf(G)ω(G) where χf(G) and ω(G) are the fractional chromatic number and clique number of G respectively. In this note, we prove that r(G) tightly captures the integrality gap of the LP relaxation with clique constraints for the Maximum Weight Independent Set (MWIS) problem. Our proof uses standard applications of multiplicative weight techniques, so it is algorithmic: Any algorithm for rounding the LP can be turned into a fractional coloring algorithm and vice versa. We discuss immediate applications of our results in approximating the fractional chromatic number of certain classes of intersection graphs.

Refining the Hierarchies of Classes of Geometric Intersection Graphs

We analyse properties to show the strict containment between some natural classes of geometric intersection graphs. For example, we show that, for each integer k≥1, the class of intersection graphs of segments with k different lengths is a strict subclass of the class of intersection graphs of segments with k+1 different lengths, and the class of outer-segment graphs is a strict subclass of the class of outer-string graphs.

An algorithm for the maximum weight independent set problem on outerstring graphs

Outerstring graphs are the intersection graphs of curves that lie inside a disk such that each curve intersects the boundary of the disk. Outerstring graphs are among the most general classes of intersection graphs studied. To date, no polynomial time algorithm is known for any of the classical graph optimization problems on outerstring graphs; in fact, most are NP-hard. It is known that there is an intersection model for any outerstring graph that consists of polygonal arcs attached to a circle. However, this representation may require an exponential number of segments relative to the size of the graph.Given an outerstring graph and an intersection model consisting of polygonal arcs with a total of N segments, we develop an algorithm that solves the Maximum Weight Independent Set problem in O(N3) time. If the polygonal arcs are restricted to single segments, then outersegment graphs result. For outersegment graphs, we solve the Maximum Weight Independent Set problem in O(n3) time where n is the number of vertices in the graph.

On the isomorphism problem for Helly circular-arc graphs

The isomorphism problem is known to be efficiently solvable for interval graphs, while for the larger class of circular-arc graphs its complexity status stays open. We consider the intermediate class of intersection graphs for families of circular arcs that satisfy the Helly property. We solve the isomorphism problem for this class in logarithmic space. If an input graph has a Helly circular-arc model, then our algorithm constructs it canonically, which means that the models constructed for isomorphic graphs are equal.

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.

Posets and VPG Graphs

We investigate the class of intersection graphs of paths on a grid (VPG graphs), and specifically the relationship between the bending number of a cocomparability graph and the poset dimension of its complement. We show that the bending number of a cocomparability graph G is at most the poset dimension of the complement of G minus one. Then, via Ramsey type arguments, we show our upper bound is best possible.

Coloring intersection graphs of x-monotone curves in the plane

A class of graphs G is ?-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 ?-bounded. As a corollary, we show that the class of intersection graphs of rays in the plane is ?-bounded, and the class of intersection graphs of unit segments in the plane is ?-bounded.