Minimum Number Of Cliques Research Articles

Product irregularity strength of graphs with small clique cover number

For a graph X without isolated vertices and without isolated edges, a product-irregular labelling ω : E(X) → {1, 2, …, s}, first defined by Anholcer in 2009, is a labelling of the edges of X such that for any two distinct vertices u and v of X the product of labels of the edges incident with u is different from the product of labels of the edges incident with v. The minimal s for which there exists a product irregular labeling is called the product irregularity strength of X and is denoted by ps(X). Clique cover number of a graph is the minimum number of cliques that partition its vertex-set. In this paper we prove that connected graphs with clique cover number 2 or 3 have the product-irregularity strength equal to 3, with some small exceptions.

Regarding Two Conjectures on Clique and Biclique Partitions

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.

Fast constructive and improvement heuristics for edge clique covering

The edge clique covering problem consists in covering the edges of a graph using the minimum number of cliques. The problem is NP-hard and even state-of-the-art heuristic methods may have prohibitive runtime or obtain low quality solutions. This paper proposes fast and better constructive and improvement heuristics for the problem. Computational experiments show the proposed heuristics can find better solutions in few seconds for most instances. Tests were run on edge clique covering, DIMACS and BHOSLIB instances.

On the kernel size of clique cover reductions for random intersection graphs

Covering all edges of a graph by a minimum number of cliques is a well known NP-hard problem. For the parameter k being the maximal number of cliques to be used, the problem becomes fixed parameter tractable. However, assuming the Exponential Time Hypothesis, there is no kernel of subexponential size in the worst-case.We study the average kernel size for random intersection graphs with n vertices, edge probability p, and clique covers of size k. We consider the well-known set of reduction rules of Gramm, Guo, Hüffner, and Niedermeier (2009) [17] and show that with high probability they reduce the graph completely if p is bounded away from 1 and k<clog⁡n for some constant c>0. This shows that for large probabilistic graph classes like random intersection graphs the expected kernel size can be substantially smaller than the known exponential worst-case bounds.

The rate of growth of the minimum clique size of graphs of given order and chromatic number

Let Q(n,c) denote the minimum clique number over graphs with n vertices and chromatic number c. We determine the rate of growth of the sequence {Q(n,⌈rn⌉)}n=1∞ for any fixed 0<r≤1. We also give a better upper bound for Q(n,⌈rn⌉).

A new characterization of perfect graphs

Let D=(V(D),A(D)) be a digraph; a kernel N of D is a set of vertices N⊆V(D) such that N is independent (for any x,y∈N, there is no arc between them) and N is absorbent (for each x∈V(D)−N, there exists an xN-arc in D). A digraph D is said to be kernel-perfect whenever each one of its induced subdigraphs has a kernel. A digraph D is oriented by sinks when every semicomplete subdigraph of D has at least one kernel. Let us recall that a graph G is perfect iff every induced subdigraph H satisfies α(H)=θ(H), where α(G) denotes the stability number of G (i.e. the maximum cardinality of an independent set of vertices of G) and θ(G) denotes the minimum number of cliques needed to cover the vertex-set of G.Let G be a graph and α=(αu)u∈V(G) a family of mutually disjoint digraphs; a sum of α over G, denoted by σ(α,G) is a digraph defined as follows. Take ⋃u∈V(G)αu, and then for each x∈V(αw) and y∈V(αv) with [w,v]∈E(G) we put at least one of the two arcs (x,y) or (y,x) in σ(α,G).The main result of this paper is the following theorem which provides a new characterization of perfect graphs.Theorem. A graphGis perfect if and only if for any familyα=(αv)v∈V(G)of mutually disjoint asymmetric kernel-perfect digraphs, any digraph constructed as a sum ofαoverG, σ(α,G)and oriented by sinks is kernel-perfect.

Minimum Clique Number, Chromatic Number, and Ramsey Numbers

Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We investigate the asymptotics of $Q(n,c)$ when $n/c$ is held constant. We show that when $n/c$ is an integer $\alpha$, $Q(n,c)$ has the same growth order as the inverse function of the Ramsey number $R(\alpha+1,t)$ (as a function of $t$). Furthermore, we show that if certain asymptotic properties of the Ramsey numbers hold, then $Q(n,c)$ is in fact asymptotically equivalent to the aforementioned inverse function. We use this fact to deduce that $Q(n,\lceil n/3 \rceil)$ is asymptotically equivalent to the inverse function of $R(4,t)$.

Minimum rank of outerplanar graphs

AbstractThe problem of finding the minimum rank over all symmetric matrices corresponding to a given graph has grown in interest recently. It is well known that the minimum rank of any graph is bounded above by the clique cover number, the minimum number of cliques needed to cover all edges of the graph. We generalize the idea of the clique cover number by defining the rank sum of a cover to be the sum of the minimum ranks of the graphs in the cover. Using this idea we obtain a combinatorial solution to the minimum rank problem for an outerplanar graph. As a consequence the minimum rank of an outerplanar graph is field independent and all outerplanar graphs have a universally optimal matrix. We also consider implications of the main result to the inverse inertia problem.

A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths that either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) it computes a clique partition having size that is guaranteed to be within (1+e) of the optimum size if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG’s is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an $O(\frac{\log^{*} n}{{\varepsilon}^{O(1)}})$ time distributed PTAS. We consider a weighted version of the clique partition problem on vertex-weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS break down. Yet, surprisingly, it admits a (2+e)-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the unweighted case for UDGs expressed in standard form.

Minimum Clique Partition in Unit Disk Graphs

The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques. Given n points in the plane, the corresponding unit disk graph (UDG) has these points as vertices, and edges connecting points at distance at most 1. MCP in UDGs is known to be NP-hard and several constant factor approximations are known, including a recent PTAS. We present two improved approximation algorithms for MCP in UDGs with a realization: (I) A polynomial time approximation scheme (PTAS) running in time $${n^{O(1/\varepsilon^2)}}$$. This improves on a previous PTAS with $${n^{O(1/\varepsilon^4)}}$$ running time by Pirwani and Salavatipour (arXiv:0904.2203v1, 2009). (II) A randomized quadratic-time algorithm with approximation ratio 2.16. This improves on a ratio 3 algorithm with O(n 2) running time by Cerioli et al. (Electron. Notes Discret. Math. 18:73–79, 2004).

Data reduction and exact algorithms for clique cover

To cover the edges of a graph with a minimum number of cliques is an NP-hard problem with many applications. For this problem we develop efficient and effective polynomial-time data reduction rules that, combined with a search tree algorithm, allow for exact problem solutions in competitive time. This is confirmed by experiments with real-world and synthetic data. Moreover, we prove the fixed-parameter tractability of covering edges by cliques.

Clique partitions of distance multigraphs

We consider the minimum number of cliques needed to partition the edge set of D ( G ) , the distance multigraph of a simple graph G. Equivalently, we seek to minimize the number of elements needed to label the vertices of a simple graph G by sets so that the distance between two vertices equals the cardinality of the intersection of their labels. We use a fractional analogue of this parameter to find lower bounds for the distance multigraphs of various classes of graphs. Some of the bounds are shown to be exact.

Packing r-Cliques in Weighted Chordal Graphs

In Hell et al. (2004), we have previously observed that, in a chordal graph G, the maximum number of independent r -cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to Kr , with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r -cliques of G. When r = 1, this says that chordal graphs have independence number equal to the clique covering number. When r = 2, this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph G, with a nonnegative weight for each r -clique in G ,w eseek a set of independent r -cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from G, and is an adaptation of an algorithm of Farber (1982). The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the r -cliques have the same weight, we simplify the algorithms reported in Hell et al. (2004), although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.

On the complexity of cell flipping in permutation diagrams and multiprocessor scheduling problems

Permutation diagrams have been used in circuit design to model a set of single point nets crossing a channel, where the minimum number of layers needed to realize the diagram equals the clique number ω ( G ) of its permutation graph, the value of which can be calculated in O ( n log n ) time. We consider a generalization of this model motivated by “standard cell” technology in which the numbers on each side of the channel are partitioned into consecutive subsequences, or cells, each of which can be left unchanged or flipped (i.e., reversed). We ask, for what choice of flippings will the resulting clique number be minimum or maximum. We show that when one side of the channel is fixed (no flipping), an optimal flipping for the other side can be found in O ( n log n ) time for the maximum clique number, and that when both sides are free this can be solved in O ( n 2 ) time. We also prove NP-completeness of finding a flipping that gives a minimum clique number, even when one side of the channel is fixed, and even when the size of the cells is restricted to be less than a small constant. Moreover, since the complement of a permutation graph is also a permutation graph, the same complexity results hold for the stable set (independence) number. In the process of the NP-completeness proof we also prove NP-completeness of a restricted variant of a scheduling problem. This new NP-completeness result may be of independent interest.

Partitioning chordal graphs into independent sets and cliques

We consider the following generalization of split graphs: A graph is said to be a ( k,ℓ)-graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k=ℓ=1.) Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by ( k,ℓ)-graphs in general. (For instance, being a ( k,0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal ( k,ℓ)-graphs. We also give an O( n( m+ n)) recognition algorithm for chordal ( k,ℓ)-graphs. When k=1, our algorithm runs in time O( m+ n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem—given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min–max property of chordal graphs: for each integer r⩾1, the maximum number of independent K r 's (i.e., of vertex disjoint subgraphs of G, each isomorphic to K r , with no edges joining two of the subgraphs) equals the minimum number of cliques of G that meet all the K r 's of G.

Asymptotic Clique Covering Ratios of Distance Graphs

Given a finite set D of positive integers, the distance graph G(Z, D) has Z as the vertex set and { ij: | i−j | ∈D } as the edge set. Given D, the asymptotic clique covering ratio is defined as S(D) =lim supn→∞ncl(n), where cl(n) is the minimum number of cliques covering any consecutive n vertices of G(Z, D). The parameter S(D) is closely related to the ratio spT(G)χ(G)of a graph G, where χ(G) and spT(G) denote, respectively, the chromatic number and the optimal span of a T -coloring of G. We prove that for any finite set D, S(D) is a rational number and can be realized by a ‘periodical’ clique covering of G(Z, D). Then we investigate the problem for which sets D the equality S(D) =ω(G(Z, D)) holds. (In general, S(D) ≤ω(G(Z, D)), where ω(G) is the clique number of G.) This problem turns out to be related to T -colorings and to fractional chromatic number and circular chromatic number of distance graphs. Through such connections, we shall show that the equality S(D) =ω(G(Z, D)) holds for many classes of distance graphs. Moreover, we raise questions regarding other such connections.

Covering the edges of a random graph by cliques

The clique cover number θ1(G) of a graph G is the minimum number of cliques required to cover the edges of graph G. In this paper we consider θ1(Gn,p), for p constant. (Recall that in the random graph Gn,p, each of the ( n 2 ) edges occurs independently with probability p). Bollobas, Erdős, Spencer and West [1] proved that whp (i.e. with probability 1-o(1) as n→∞) (1− o(1))n 4(log2 n) 2 ≤ θ1(Gn,.5) ≤ cn ln lnn (lnn)2 .

A neural network parallel algorithm for clique vertex-partition problems

A parallel algorithm based on a neural network model for solving clique vertex-partition problems in arbitrary non-directed graphs is presented in this paper. A clique of a graph G = (V, E) with a set of vertices V and a set of edges E is a complete subgraph of G where any pair of vertices is connected with an edge. A clique vertex-partition problem of a graph G is to partition every vertex in V into a set of disjointed cliques of G. The clique vertex-partition problem with the minimum number of cliques in an arbitrary graph is known to be NP-complete. The algorithm requires nm processing elements for the n vertex m partition problem. A total of 10 different problems with 8 vertex to 300 vertex graphs were examined where the algorithm found a solution in nearly constant time. The circuit diagram of the neural network model is also proposed in this paper.

Graph representation of logic functions to design two-level MOS networks

The design problem of two-level MOS networks with a minimum number of cells is reduced to the covering problem of graphs with a minimum number of cliques. When every node of the graph corresponds to a minterm, an algorithm which generates an MOS network with a minimum number of cells is presented. The time complexity of this algorithm is O(M)2, where M is the number of minterms. When every node of the graph corresponds to a prime implicant, a procedure which generates near optimum solutions by selecting prime implicants with special properties is presented. About 2 percent of designed networks for four-variable functions are not minimal ones.

Perfect Graphs and Orthogonally Convex Covers

We consider the problem of covering simple orthogonal polygons with convex orthogonal polygons. In the case of horizontally or vertically convex polygons we show that the polygon covering problem can be reduced to the problem of covering a permutation graph with minimum number of cliques. In general, orthogonal polygons can have concavities (dents) with four possible orientations. In the case where the polygon has three dent orientations, we show that the polygon covering problem can be reduced to the problem of covering a weakly triangulated graph with a minimum number of cliques. Since weakly triangulated graphs are perfect, we obtain the following duality relationship: the minimum number of orthogonally convex polygons needed to cover an orthogonal polygon P with at most three dent orientations is equal to the maximum number of points of P, no two of which can be contained together in an orthogonally convex covering polygon. Finally, we show that in th case of orthogonal polygons with all four dent orientations, the above duality relationship fails to hold.