Fully dynamic representations of interval graphs

Interval graph representation with given interval and intersection lengths

Algorithmic aspects of switch cographs

A fully dynamic algorithm for modular decomposition and recognition of cographs

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: MINIMUM VERTEX COVER, MINIMUM DOMINATING SET, and MAXIMUM CLIQUE. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multiple-interval graph. For MINIMUM VERTEX COVER, we give a (2--1/t)-approximation algorithm which equals the best known ratio for 2t--1 bounded degree graphs. Since these graphs are known to be included in multiple-interval graphs with t intervals associated to each vertex, this ratio is in some sense tight. Following this, we give a t2-approximation algorithm for MINIMUM DOMINATING SET which adapts well to more general variants of the problem. We then proceed to prove that MAXIMUM CLIQUE is NP-complete for the case of t = 3, and provide a (t2--t+1)/2-approximation algorithm for the problem, using recent bounds proven for the so-called transversal number of t-interval families.

Theinterval count problemdetermines the smallest number of interval lengths needed in order to represent an interval model of a given interval graph or interval order. Despite the large number of studies about interval graphs and interval orders, surprisingly only a few results on the interval count problem are known. In this work, we provide a short survey about the interval count and related problems. a graph and the number of its maximal cliques.

A polynomial algorithm for the k-cluster problem on the interval graphs

Counting spanning trees using modular decomposition

Klavík et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals \(\mathfrak{L}_v\) and \(\mathfrak{R}_v\) called bounds. We ask whether there exists a bounded representation in which each interval \(\mathfrak{I}_v\) has its left endpoint in \(\mathfrak{L}_v\) and its right endpoint in \(\mathfrak{R}_v\). We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs.Robert’s Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly, the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klavík et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently.The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.

New sequential and parallel algorithms for interval graph recognition

Finding depth-first-search (DFS) trees of graphs is one of the fundamental problems in graph theory which has many practical applications. However, there exists no NC parallel algorithm for this problem in general graphs. NC parallel algorithms for finding DFS trees for the interval graphs and circular-arcs graphs are presented. These algorithms take O(log n) time using kappa n processors on the EREW model, where kappa is the number of cliques in a graph of n nodes. >

An interval graph G is homogeneously representable if for every vertex v of G there exists an interval representation of G with v corresponding to an end interval. We show that the homogeneous representation of interval graphs is rooted in a deeper property of a class of graphs that we characterize by forbidden configurations.

In this paper we address the problem of designing O(n) space representations for permutation and interval graphs that provide the neighborhood of any vertex in O(d) time, where d is its degree. To that purpose, we introduce a new parameter, called linearity, that would solve the problem if bounded for the two classes. Surprisingly, we show that it is not. Nevertheless, we design representations with the desired property for the two classes, and we implement the Breadth-First Search algorithm in O(n) time for permutation graphs; thereby lowering the complexity of All Pairs Shortest Paths and Single Source Shortest Path problems for the class.

Many {\cal NP}-hard graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for diverse optimization problems. To apply a tree decomposition approach, the treewidth of the graph has to be determined, independently of the application at hand. Although for fixed $k$, linear time algorithms exist to solve the decision problem ``treewidth $\leq k$'', their practical use is very limited. The computational tractability of treewidth has been rarely studied so far. In this paper, we compare four heuristics and two lower bounds for instances from applications such as the frequency assignment problem and the vertex coloring problem. Three of the heuristics are based on well-known algorithms to recognize triangulated graphs. The fourth heuristic recursively improves a tree decomposition by the computation of minimal separating vertex sets in subgraphs. Lower bounds can be computed from maximal cliques and the minimum degree of induced subgraphs. A computational analysis shows that the treewidth of several graphs can be identified by these methods. For other graphs, however, more sophisticated techniques are necessary.

Graph theory offers a convenient and highly attractive approach to various tasks of pattern recognition. Provided there is a graph representation of the object in question (e.g. a chemical structure or protein fold), the recognition procedure is reduced to the problem of common subgraph isomorphism (CSI). Complexity of this problem shows combinatorial dependence on the size of input graphs, which in many practical cases makes the approach computationally intractable. Among the optimal algorithms for CSI, the leading place in practice belongs to algorithms based on maximal clique detection in the association graph. Backtracking algorithms for CSI, first developed two decades ago, are rarely used. We propose an improved backtracking algorithm for CSI, which differs from its predecessors by better search strategy and is therefore more efficient. We found that the new algorithm outperforms the traditional maximal clique approach by orders of magnitude in computational time. Copyright © 2004 John Wiley & Sons, Ltd.

We prove a characterization of all polynomial-time computable queries on the class of interval graphs by sentences of fixed-point logic with counting. More precisely, it is shown that on the class of unordered interval graphs, any query is polynomial-time computable if and only if it is definable in fixed-point logic with counting. This result is one of the first establishing the capturing of polynomial time on a graph class which is defined by forbidden induced subgraphs. For this, we define a canonical form of interval graphs using a type of modular decomposition, which is different from the method of tree decomposition that is used in most known capturing results for other graph classes, specifically those defined by forbidden minors. The method might also be of independent interest for its conceptual simplicity. Furthermore, it is shown that fixed-point logic with counting is not expressive enough to capture polynomial time on the classes of chordal graphs or incomparability graphs.