Clique-width of full bubble model graphs

Journal of the Brazilian Computer Society, 2011

The interval count problem determines 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.

Acta Cybernetica

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. Here, we propose a family of representations for the graph G, which yield the well-known upper bound ⌈1 2(d+1)⌉, where d is the maximum degree of G. The extremal graphs for even d are also described, and the upper bound on the interval number in terms of the number of edges of G is improved.

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.

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.

Discrete Applied Mathematics

Given an interval graph G, the interval count problem is that of computing the minimum number IC(G) of interval lengths needed to represent G. Although the problem of deciding whether IC(G)=1 is equivalent to that of recognizing unit-interval graphs, which is a well-known problem having several efficient recognition approaches, very little is known about deciding efficiently whether IC(G)=k for fixed k≥2. We provide efficient computations of the interval count of generalizations of threshold graphs.

2000

position, but it is allowed to move, provided that it remains completely contained into its window. Once a position has been fixed for all the intervals, the graph becomes an usual interval graph. In this paper we address the problem of finding the position of the intervals, which minimizes or maximizes some classical measures of the graph, such as clique number, stability number, chromatic number, clique cover number. We mainly focus on complexity aspects, bounds and solution algorithms. Some problems are solvable in polynomial time, others are proved to be NP-hard. Moreover some subclasses of SIG's ,for which exist polynomial algorithms exist, are characterized. Many practical applications can be reduced to problems on SIG's, and SIG's seem to be an interesting modeling framework.

Algorithmica, 2013

Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t ≥ 3 and polynomial-time solvable when t = 1. The problem is also known to be NP-complete in t-track graphs when t ≥ 4 and polynomialtime solvable when t ≤ 2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APXcomplete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t. 2 Preliminaries Consider a circle C of length l with a distinguished point O. The coordinate of a point p ∈ C is the length of the arc going clockwise from O to p. Given two reals p and q, [p, q] is the arc of C going clockwise from the point with coordinate p to the one with coordinate q. In the following, coordinates are understood modulo l.

The interval count problem is that of determining the smallest num-ber of interval lengths required 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, few results on the interval count problem exist in fact. We provide a short survey about the interval count and related problems.

2010

Clique-width is an important graph parameter whose computation is NP-hard. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any non-trivial graph class. Results so far indicate that proper interval graphs constitute the first interesting graph class on which we might have hope to compute clique-width, or at least its linear variant linear clique-width, in polynomial time. In TAMC 2009, a polynomialtime algorithm for computing linear clique-width on a subclass of proper interval graphs was given. In this paper, we present a polynomial-time algorithm for a larger subclass of proper interval graphs that approximates the clique-width within an additive factor 3. Previously known upper bounds on clique-width result in arbitrarily large difference from the actual clique-width when applied on this class. Our results contribute toward the goal of eventually obtaining a polynomial-time exact algorithm for clique-width on proper interval graphs.

Lecture Notes in Computer Science, 2009

We investigate connected proper interval graphs without vertex labels. We first give the number of connected proper interval graphs of n vertices. Using this result, a simple algorithm that generates a connected proper interval graph uniformly at random up to isomorphism is presented. Finally an enumeration algorithm of connected proper interval graphs is proposed. The algorithm is based on reverse search, and it outputs each connected proper interval graph in O(1) time.