Folding graphs

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011

A d-dimensional Keller graph has vertices which are numbered with each of the 4 d possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a "high speed computer the size of a major galaxy". This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124.

Matemática Contemporânea, 2010

A complete set of a graph G is a subset of V G whose elements are pairwise adjacent. A clique is a maximal complete set. The clique graph of G, denoted by K(G), is the intersection graph of the family of cliques of G. The clique graph recognition problem asks whether a given graph is a clique graph. This problem was classified recently as NP-complete after being open for 30 years. The complexity of this decision problem is open for very structured and well studied classes of graphs such as planar graphs and chordal graphs. We propose the study of split clique graphs.

Theoretical Computer Science, 2013

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C(G) the clique family of G. The clique graph of G, denoted by K (G), is the intersection graph of C(G). Say that G is a clique graph if there exists a graph H such that G = K (H). The clique graph recognition problem, a longstanding open question posed in 1971, asks whether a given graph is a clique graph and it was recently proved to be NP-complete even for a graph G with maximum degree 14 and maximum clique size 12. Hence, if P 6 = NP, the study of graph classes where the problem can be proved to be polynomial, or of more restricted graph classes where the problem remains NP-complete is justified. We present a proof that given a split graph G = (V , E) with partition (K , S) for V , where K is a complete set and S is a stable set, deciding whether there is a graph H such that G is the clique graph of H is NP-complete. As a byproduct, we prove that determining whether a given set family admits a spanning family satisfying the Helly property is NP-complete. Our result is optimum in the sense that each vertex of the independent set of our split instance has degree at most 3, whereas when each vertex of the independent set has degree at most 2 the problem is polynomial, since it is reduced to the problem of checking whether the clique family of the graph satisfies the Helly property. Additionally, we show three split graph subclasses for which the problem is polynomially solvable: the subclass where each vertex of S has a private neighbor, the subclass where |S| 6 3, and the subclass where |K | 6 4.

2010

A graph is a (planar, Kh)-graph if a collection of disjoint clusters can be identified such that the subgraph induced by each cluster is an h-clique and collapsing all clusters yields a planar graph. Recognizing (planar, Kh)-graphs is a special instance of the more general problem of recognizing (X ,Y)-graphs, where X and Y are two chosen families of graphs. This model, together with the (X ,Y)-graph terminology, was introduced in [3] and generalized in [1, 2] to support hybrid visualization. In particular, if the clusters are requested to be a partition of the set of the vertices of the input graph G, as it is in [3], then G is called a strong (X ,Y)-graph, otherwise G is a weak (X ,Y)-graph or, simply, an (X ,Y)-graph. In this paper we address (planar, Kh)-recognition in the weak model, and show that this problem is NP-complete if h ≥ 5. This result parallels the analogous result for the strong model [10, 9]. We remark that allowing the contraction of any clique of size greater th...

Maximum Clique Problem(MCP) is one of the 21 original NP--complete problems enumerated by Karp in 1972. In recent years a large number of exact methods to solve MCP have been appeared(Babel, Wood, Kumlander, Fahle, Li, Tomita and etc). Most of them are branch and bound algorithms that use branching rule introduced by Balas and Yu and based on coloring heuristics to establish an upper bound on the clique number. They differ from each other primarily in vertex preordering and vertex coloring methods. Current methods of worst case running time analysis for branch and bound algorithms do not allow to provide tight upper bounds. This motivates the study of lower bounds for such algorithms. We prove 2^(n\5) lower bound for group of MCP algorithms based on usage of coloring heuristics.

2010

The main focus of this thesis is to evaluate kr(n, δ), the minimal number of r-cliques in graphs with n vertices and minimum degree δ. A fundamental result in Graph Theory states that a triangle-free graph of order n has at most n/4 edges. Hence, a triangle-free graph has minimum degree at most n/2, so if k3(n, δ) = 0 then δ ≤ n/2. For n/2 ≤ δ ≤ 4n/5, I have evaluated kr(n, δ) and determined the structures of the extremal graphs. For δ ≥ 4n/5, I give a conjecture on kr(n, δ), as well as the structures of these extremal graphs. Moreover, I have proved various partial results that support this conjecture. Let k r (n, δ) be the analogous version of kr(n, δ) for regular graphs. Notice that there exist n and δ such that kr(n, δ) = 0 but k reg r (n, δ) > 0. For example, a theorem of Andrasfai, Erdős and Sos states that any triangle-free graph of order n with minimum degree greater than 2n/5 must be bipartite. Hence k3(n, bn/2c) = 0 but k 3 (n, bn/2c) > 0 for n odd. I have evaluated ...

2019

A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We show here that the potential maximal cliques of a graph can be generated in polynomial time in the number of minimal separators of the graph. Thus, the treewidth and the minimum fill-in are polynomially tractable for all classes of graphs with a polynomial number of minimal separators.

Lecture Notes in Computer Science, 2016

We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H.

Electronic Proceedings in Theoretical Computer Science

We address the problem of enumerating all maximal clique-partitions of an undirected graph and present an algorithm based on the observation that every maximal clique-partition can be produced from the maximal clique-cover of the graph by assigning the vertices shared among maximal cliques, to belong to only one clique. This simple algorithm has the following drawbacks: (1) the search space is very large; (2) it finds some clique-partitions which are not maximal; and (3) some cliquepartitions are found more than once. We propose two criteria to avoid these drawbacks. The outcome is an algorithm that explores a much smaller search space and guarantees that every maximal cliquepartition is computed only once. The algorithm can be used in problems such as anti-unification with proximity relations or in resource allocation tasks when one looks for several alternative ways to allocate resources.

In this text we provide an algorithm giving an upper bound on the size of the maximum clique in a graph. The algorithm uses induced subgraphs and rules designed to decrease a set of properties as much as possible. Time and memory requirements are O(n^5) and O(n^3) respectively. Evaluation on random graphs suggests that the upper bounds depend linearly on the graph orders, and the slope depends on the probability that certain edges exist in a random graph. This paper also provides an introduction to variants of the clique problem and discusses a memory ecient data structure representing undirected graphs. 1. Introduction Many different algorithms can be constructed to solve one specic problem. Some might be more ecient than others. An algorithm's eciency is determined by its time and memory consumption. Often these measures are analyzed by theoretical means, but when an algorithm's behavior is too complex to understand, simulations and statistics might come in handy. Computat...