On split clique graphs

Annals of Operations Research, 2002

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A cliqueindependent set is a subset of pairwise disjoint cliques of G. Denote by τ C (G) and α C (G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τ C (H ) = α C (H ), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters τ C (G) and α C (G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference τ C (G) − α C (G) is arbitrarily large.

Discrete Applied Mathematics, 1996

Suppose G=( V,E) is a graph in which each maximal clique Ci is associated with an integer ri, where 0 < ri < ICi 1. The generalized clique transversal problem is to determine the minimum cardinality of a subset D of V such that ID n Cij >ri for every maximal clique Ci of G. The problem includes the clique-transversal problem, the i, 1 clique-cover problem, and for perfect graphs, the maximum q-colorable subgraph problems as special cases. This paper gives complexity results for the problem on subclasses of chordal graphs, e.g., strongly chordal graphs, k-trees, split graphs, and undirected path graphs.

2020

A clique in a graph is a set of its vertices which are all connected to each other. A clique partition of a graph is a set of its disjoint cliques, which cover all the vertices of the graph. A partition is maximal, if no two cliques in it can be joined into a bigger one. Computing one maximal clique partition is a well-studied problem and several algorithms exist for it. In this paper, we consider the problem of computing all maximal clique partitions. We start from all maximal cliques of a graph and proceed by trying to make them disjoint, assigning shared vertices to one of the cliques they belong to. The challenge is to compute only the needed solutions. Our algorithm returns only maximal partitions, and each of them is computed once. Tests that detect failing branches early, guarantee that no false answer is first computed and then discarded. These properties make the branches of the algorithm’s search space independent from each other. Therefore, the process is easily paralleli...

Journal of Graph Theory, 2005

A biclique cutset is a cutset that induces the disjoint union of two cliques. A hole is an induced cycle with at least five vertices. A graph is biclique separable if it has no holes and each induced subgraph that is not a clique contains a clique cutset or a biclique cutset. The class of biclique separable graphs contains several well-studied graph classes, including triangulated graphs. We prove that for the class of biclique separable graphs, the recognition problem, the vertex coloring problem, and the clique problem can be solved efficiently. Our algorithms also yield a proof that biclique separable graphs are perfect. Our coloring algorithm is actually more general and can be applied to graphs that can be decomposed via a special type of biclique cutset. Our algorithms are based on structural results on biclique separable graphs developed in this paper.

Lecture Notes in Computer Science, 1988

We design a fast parallel algorithm for determining all maximal cliques (maximal independent sets) in an arbitrary graph, working in O(log 3 (nM)) parallel time and O(M 6 n 2 ) processors on a CREW-PRAM, where n is the number of vertices and M the number of maximal cliques. It entails the existence of deterministic NC-algorithms for several important graph classes with a polynomially bounded number of maximal cliques (maximal independent sets) in the number of vertices. Our result surprisingly generalizes the recent fast NC-algorithms of NNS] and DK 1] for computing all maximal cliques on chordal graphs to the arbitrary classes with polynomially many maximal cliques. Examples of these important classes of graphs besides chordal and strongly chordal graphs NNS], DK] are circle and circular graphs Go], GHS], K 4 ne graphs, circular arc graphs, expander graphs, and edge graphs Ga]. They arise in a number of applications Ga], TIAS], MC], GMS].

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...

Discrete Mathematics, 1985

We consider the problem of decomposing a graph by means of clique separators, by which we mean finding cliques (complete graphs) whose removal disconnects the graph. We give an O(nm)-time algorithm for finding a decomposition of an n-vertex, m-edge graph. We describe how such a decomposition can be used in divide-and-conquer algorithms for various graph problems, such as graph coloring and finding maximum independent sets. We survey classes of graphs for which such divide-and-conquer algorithms are especially useful.

Discrete Applied Mathematics, 2006

A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph. In the present paper, we propose polynomial time algorithms for finding the maximum cardinality and weight of a clique-independent set of a 3K 2 -free CA graph. Also, we apply the algorithms to the special case of an HCA graph. The complexity of the proposed algorithm for the cardinality problem in HCA graphs is O(n). This represents an improvement over the existing algorithm by Guruswami and Pandu Rangan, whose complexity is O(n 2 ). These algorithms suppose that an HCA model of the graph is given.

Discrete Applied Mathematics, 2012

Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently. One of the major open questions regarding clique-width is whether graphs of cliquewidth at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n 2 m)) to recognize graphs of clique-width at most 3.

Discrete Mathematics, 2002

The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlike-threshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge clique graph is that the sizes of all maximal cliques and intersections of maximal cliques ought to be triangular numbers. We show that this condition is also su cient for starlike-threshold graphs.