To cover the edges of a graph with a minimum number of cliques is an NP-complete problem with many applications. The state-of-the-art solving algorithm is a polynomial-time heuristic from the 1970’s. We present an improvement of this heuristic. Our main contribution, however, is the development of efficient and effective polynomial-time data reduction rules that, combined with a search tree alg...

In this paper those graphs are studied for which a so-called strong ordering of the vertex set exists. This class of graphs, called strongly orderable graphs, generalizes the strongly chordal graphs and the chordal bipartite graphs in a quite natural way. We consider two characteristic elimination orderings for strongly orderable graphs, one on the vertex set and the second on the edge set, and...

The densest k-subgraph problem is a generalization of the maximum clique problem, in which we are given a graph G and a positive integer k, and we search among the subsets of k vertices of G one inducing a maximum number of edges. In this paper, we present algorithms for finding exact solutions of densest k-subgraph improving the standard exponential time complexity of O∗(2n) and using polynomi...

For our purposes, graphs have no loops or multiple edges. Aside from this, our terminology follows that of Bondy and Murty [1]. A clique is a complete subgraph. A clique partition of a graph G is a family ~ of cliques of G such that each edge of G is in precisely one member of q¢. The clique partition number, cp(G), is the minimum of the cardinalities of the clique partitions of G. Let To denot...

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arb...

In social network analysis, a k-clique is a relaxed clique, i.e., a kclique is a quasi-complete sub-graph. A k-clique in a graph is a sub-graph where the distance between any two vertices is no greater than k. The visualization of a small number of vertices can be easily performed in a graph. However, when the number of vertices and edges increases the visualization becomes incomprehensible. In...

Two sets are non-crossing if they are disjoint or one contains the other. The noncrossing graph NCn is the graph whose vertex set is the set of nonempty subsets of [n] = {1, . . . , n} with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover nu...

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number ...

Vertex colouring is a well-known problem in combinatorial optimisation, whose alternative integer programming formulations have recently attracted considerable attention. This paper briefly surveys seven known formulations of vertex colouring and introduces a new formulation of vertex colouring using a suitable clique partition of the graph. This formulation is applicable in timetabling applica...

A t-chrome cover is a cover of a weighted graph by vertex sets that induce t-colourable subgraphs. We investigate the problem of determining the minimal number of sets needed for such a cover, and give conditions under which this number is directly derived from the clique number and the maximum weight of the graph. The problem of minimal t-chrome covers is relevant to frequency assignment in ce...