On the structure of certain intersection graphs (original) (raw)

Interval bigraphs and circular arc graphs

Journal of Graph Theory, 2004

We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two, which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidaltriple-free graphs, permutation graphs, and co-comparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of Mü ller. ß

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.

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.

Obstructions to chordal circular-arc graphs of small independence number

Electronic Notes in Discrete Mathematics, 2013

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [11]. We show that a circular-arc graph cannot have a blocking quadruple. We also observe that the absence of blocking quadruples is not in general sufficient to guarantee that a graph is a circular-arc graph. Nonetheless, it can be shown to be sufficient for some special classes of graphs, such as those investigated in [2]. In this note, we focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples. Our contribution is twofold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [11]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circulararc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.

On the interval number of special 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.

Balancedness of some subclasses of circular-arc graphs

Electronic Notes in Discrete Mathematics, 2010

A graph is balanced if its clique-vertex incidence matrix is balanced, i.e., it does not contain a square submatrix of odd order with exactly two ones per row and per column. Interval graphs, obtained as intersection graphs of intervals of a line, are well-known examples of balanced graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. Circular-arc graphs generalize interval graphs, but are not balanced in general. In this work we characterize balanced graphs by minimal forbidden induced subgraphs restricted to graphs that belong to some classes of circular-arc graphs.

A new representation of proper interval graphs with an application to clique-width

Electronic Notes in Discrete Mathematics, 2009

We introduce a new representation of proper interval graphs that can be computed in linear time and stored in O(n) space. This representation is a 2-dimensional vertex partition. It is particularly interesting with respect to clique-width. Based on this representation, we prove new upper bounds on the clique-width of proper interval graphs.

Intersection Graphs and the Clique Operator

Graphs and Combinatorics, 2001

Let P be a class of ®nite families of ®nite sets that satisfy a property P. We call XP the class of intersection graphs of families in P and CliqueP the class of graphs whose family of cliques is in P. We prove that a graph G is in XP if and only if there is a family of complete sets of G which covers all edges of G and whose dual family is in P. This result generalizes that of Gavril for circular-arc graphs and conduces those of Fulkerson-Gross, Gavril and Monma-Wei for interval graphs, chordal graphs, UV , DV and RDV graphs. Moreover, it leads to the characterization of Helly-graphs and dually chordal graphs as classes of intersection graphs. We prove that if P is closed under reductions, then CliqueP XP Ã H (P Ã Class of dual families of families in P). We ®nd sucient conditions for the Clique Operator, K, to map XP into XP Ã. These results generalize several known results for particular classes of intersection graphs. Furthermore, they lead to the Roberts-Spencer characterization for the image of K and the Bandelt-Prisner result on K-®xed classes.

Maximal-clique partitions of interval graphs

Journal of the Australian Mathematical Society, 1988

It is shown that if an interval graph possesses a maximal-clique partition then its clique covering and clique partition numbers are equal, and equal to the maximal-clique partition number. Moreover an interval graph has such a partition if and only if all its maximal cliques are edge-disjoint.

Intersection Graphs: An Introduction

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.