Intersection Graphs: An Introduction
Madhumangal Pal
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Abstract
Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation, trapezoid, chordal, disk, circle graphs are more important. In this article, a brief introduction of each of these intersection graphs is given. Some basic properties and algorithmic status of few problems on these graphs are cited. This article will help to the beginners to start work in this direction. Since the article contains a lot of information in a compact form it is also useful for the expert researchers too.
Related papers
Algorithmic Aspects of the Intersection and Overlap Numbers of a Graph
Lecture Notes in Computer Science, 2012
The intersection number of a graph G is the minimum size of a ground set S such that G is an intersection graph of some family of subsets F ⊆ 2 S . The overlap number of G is defined similarly, except that G is required to be an overlap graph of F. Computing the overlap number of a graph has been stated as an open problem in [B. In this paper we show two algorithmic aspects concerning both these graph invariants. On the one hand, we show that the corresponding optimization problems associated with these numbers are both APX-hard, where for the intersection number our results hold even for biconnected graphs of maximum degree 7, strengthening the previously known hardness result. On the other hand, we show that the recognition problem for any specific intersection graph class (e.g. interval, unit disc, string, ...) is easy when restricted to graphs of fixed bounded intersection or overlap number.
On the structure of certain intersection graphs
Information Processing Letters, 2006
Recently Gavril introduced a new class of intersection graphs called interval-filament graphs. These include co-comparability graphs and polygon-circle graphs (the intersection graphs of polygons inscribed in a circle), which include circular-arc graphs (the intersection graphs of arcs of a circle), circle graphs (the intersection graphs of chords of a circle), chordal graphs, and outerplanar graphs. We give a structural property of polygon-circle graphs. We prove a bound on the clique-covering number for interval-filament graphs in terms of the size of a largest independent set of nodes in the graph. We prove that complements of interval-filament graphs are 4-divisible in the sense of Hoàng and McDiarmid. Similar results are obtained for complements of other intersection graphs introduced by Gavril.
Tractabilities and Intractabilities on Geometric Intersection Graphs
Algorithms, 2013
A graph is said to be an intersection graph if there is a set of objects such that each vertex corresponds to an object and two vertices are adjacent if and only if the corresponding objects have a nonempty intersection. There are several natural graph classes that have geometric intersection representations. The geometric representations sometimes help to prove tractability/intractability of problems on graph classes. In this paper, we show some results proved by using geometric representations.
The Balanced Connected Subgraph Problem for Geometric Intersection Graphs
SSRN Electronic Journal, 2022
We study the Balanced Connected Subgraph (shortly, BCS) problem on geometric intersection graphs such as interval, circulararc, permutation, unit-disk, outer-string graphs, etc. Given a vertexcolored graph G = (V, E), where each vertex in V is colored with either "red " or "blue", the BCS problem seeks a maximum cardinality induced connected subgraph H of G such that H is color-balanced , i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithm for the Steiner Tree problem on both the interval graphs and circular arc graphs, that is used as a subroutine for solving BCS problem on same graph classes. Finally, we present a FPT algorithm for the BCS problem on general graphs.
Edge Intersection Graphs of L-Shaped Paths in Grids
Electronic Notes in Discrete Mathematics, 2013
In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: , , , , and we consider zero bend paths (i.e., | and-) to be degenerate 's. These graphs, called B 1-EPG graphs, were first introduced by Golumbic et al (2009). We consider the natural subclasses of B 1-EPG formed by the subsets of the four single bend shapes (i.e., { }, { , }, { , }, and { , , }) and we denote the classes by [ ], [ , ], [ , ], and [ , , ] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [ ] [ , ], [ , ] [ , , ] B 1-EPG and [ , ] is incomparable with [ , ]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split ∩ [ ].
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers, 2000
An interesting class of "geometric intersection problems" calls for dealing with the pairwise intersections among a set of N objects in the plane, These problems arise in many applications such as printed circuit design, architectural data bases, and computer graphics. Shamos and Hoey have described a number of algorithms for detecting whether any two objects in a planar set intersect. In this paper we extend their work by giving algorithms that count the number of such intersections and algorithms that report all such intersections.
New Results on Intersection Query Problems
The Computer Journal, 1997
We present simple algorithms for three problems belonging to the class of intersection query problems. The "rst algorithm deals with the static rectangle enclosure problem and can easily be extended to d dimensions, the second algorithm copes with the generalized c-oriented polygon intersection searching problem in two dimensions, while the third solves the static 2-dimensional dominance searching problem with respect to a set of obstacles. All algorithms are simple, are based on persistence and improve previous bounds. Also, as a corollary of the "rst algorithm, we present a result for the static d-dimensional range searching problem.
Intersection graphs of paths in a tree
Journal of Combinatorial Theory, Series B, 1986
The intersection graph for a family of sets is obtained by associating each set with a vertex of the graph and joining two vertices by an edge exactly when their corresponding sets have a nonempty intersection. Intersection graphs arise naturally in many contexts, such as scheduling conflicting events, and have been widely studied. We present a unilied framework for studying several classes of intersection graphs arising from families of paths in a tree. Four distinct classes of graphs arise by considering paths to be the sets of vertices or the edges making up the path, and by allowing the underlying tree to be undirected or directed; in the latter case only directed paths are allowed. Two further classes are obtained by requiring the directed tree to be rooted. We introduce other classes of graphs as well. The rooted directed vertex path graphs have been studied by Gavril: the vertex path graphs have been studied by Gavril and Renz; the edge path graphs have been studied by Golumbic and Jamison, Lobb, Syslo. and Tarjan. The main results are a characterization of these graphs in terms of their "clique tree"representations and a unified recognition algorithm. The algorithm repeatedly separates an arbitrary graph by a (maximal) clique separator, checks the form of the resultant nondecomposable "atoms," and finally checks that each separation step is valid. In all cases, the first two steps can be performed in polynomial time. In all but one case, the final step is based on a certain two-coloring condition and so can be done efficiently; in the other case the recognition problem can be shown to be NP-complete since a certain three-coloring condition is needed. The strength of this unified approach is that it clarifies and unities virtually all of the important known results for these graphs and provides substantial new results as well. For example, the exact intersecting relationships among these graphs. and between these graphs and chordal and perfect graphs fall out easily as corollaries. A number of other results, such as bounds on the number of (maximal) cliques, related optimization problems on these graphs, etc., are presented along with open problems.
Spanners for Geometric Intersection Graphs
Lecture Notes in Computer Science
Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R k , a (1 + ǫ)-spanner with O(nǫ −k+1) edges is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.
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Related papers
Structure of intersection graphs
Indonesian Journal of Combinatorics, 2021
Let G be a finite group and let N be a fixed normal subgroup of G. In this paper, a new kind of graph on G, namely the intersection graph is defined and studied. We use Γ in G (N) to denote this graph, with its vertices are all normal subgroups of G and two distinct vertices are adjacent if their intersection in N. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of Γ in G ({e}) for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups.
On a class of intersection graphs
Given a directed graph D = (V,A) we define its intersection graph I(D) = (A,E) to be the graph having A as a node-set and two nodes of I(D) are adjacent if their corresponding arcs share a common node that is the tail of at least one of these arcs. We call these graphs facility location graphs since they arise from the classical uncapacitated facility location problem. In this paper we show that facility location graphs are hard to recognize and they are easy to recognize when the graph is triangle-free. We also determine the complexity of the vertex coloring, the stable set and the facility location problems on that class.
Finding intersection models: From chordal to Helly circular-arc graphs
Discrete Mathematics, 2012
Every chordal graph G admits a representation as the intersection graph of a family of subtrees of a tree. A classic way of finding such an intersection model is to look for a maximum spanning tree of the valuated clique graph of G. Similar techniques have been applied to find intersection models of chordal graph subclasses as interval graphs and path graphs. In this work, we extend those methods to be applied beyond chordal graphs: we prove that a graph G can be represented as the intersection of a Helly separating family of graphs belonging to a given class if and only if there exists a spanning subgraph of the clique graph of G satisfying a particular condition. Moreover, such a spanning subgraph is characterized by its weight in the valuated clique graph of G. The specific case of Helly circular-arc graphs is treated. We show that the canonical intersection models of those graphs correspond to the maximum spanning cycles of the valuated clique graph.
An optimal algorithm for finding segments intersections
Proceedings of the eleventh annual symposium on Computational geometry - SCG '95, 1995
This paper deals with a new deterministic algorithm for finding intersecting pairs from a given set of N segments in the plane. The algorithm is asymptotically optimal and has time and space complexity O(AJ log N+ K) and 0(IV) respectively, where K is the number of intersecting pairs. The algorithm may be used for finding intersections not only line segments but also curve segments.
Balanced Connected Subgraph Problem in Geometric Intersection Graphs
Combinatorial Optimization and Applications, 2019
We study the Balanced Connected Subgraph (shortly, BCS) problem on geometric intersection graphs such as interval, circulararc, permutation, unit-disk, outer-string graphs, etc. Given a vertexcolored graph G = (V, E), where each vertex in V is colored with either "red " or "blue", the BCS problem seeks a maximum cardinality induced connected subgraph H of G such that H is color-balanced , i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithm for the Steiner Tree problem on both the interval graphs and circular arc graphs, that is used as a subroutine for solving BCS problem on same graph classes. Finally, we present a FPT algorithm for the BCS problem on general graphs.
A Search-Free Intersection Algorithm
2008
where k is the number of sorted sets and n is the total elements of k sorted sets.
Powers of geometric intersection graphs and dispersion algorithms
Discrete applied …, 2003
We study powers of certain geometric intersection graphs: interval graphs, m-trapezoid graphs and circular-arc graphs. We define the pseudo product, (G, G ) → G * G , of two graphs G and G on the same set of vertices, and show that G * G is contained in one of the three classes of graphs mentioned here above, if both G and G are also in that class and fulfill certain conditions. This gives a new proof of the fact that these classes are closed under taking power; more importantly, we get efficient methods for computing the representation for G k if k ≥ 1 is an integer and G belongs to one of these classes, with a given representation sorted by endpoints. We then use these results to give efficient algorithms for the k-independent set, dispersion and weighted dispersion problem on these classes of graphs, provided that their geometric representations are given.
On Intersection Graphs of Segments with Prescribed Slopes
Lecture Notes in Computer Science, 2002
We study intersection graphs of segments with prescribed slopes in the plane. A sufficient and necessary condition on tuples of slopes in order to define the same class of graphs is presented for both the possibilities that the parallel segments can or cannot overlap. Classes of intersection graphs of segments with four slopes are fully described; in particular, we find an infinite set of quadruples of slopes which define mutually distinct classes of intersection graphs of segments with those slopes.
Intersection representations of graphs by arcs
Pacific Journal of Mathematics, 1970
General results on tolerance intersection graphs
Journal of Graph Theory, 1991
In this paper, general results are presented that are related to +tolerance intersection graphs previously defined by Jacobson, McMorris, and Mulder. For example, it is shown that all graphs are &tolerance intersection graphs for all 4, yet for "nice" 4, almost no graphs are &tolerance interval graphs. Additional results about representation of trees are given.
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