Lhouari Nourine

Enumeration aspects of maximal cliques and bicliques

We present a general framework to study enumeration algorithms for maximal cliques and maximal bicliques of a graph. Given a graph G, we introduce the notion of the transition graph T(G) whose vertices are maximal cliques of G and arcs are transitions between cliques. We show that T(G) is a strongly connected graph and characterize a rooted cover tree of T(G) which appears implicitly in [D.S. Johnson, M. Yannakakis, C.H. Papadimitriou, On generating all maximal independent sets, Information Processing Letters 27 (1988) 119-123; S. Tsukiyama, M. Ide, M. Aiyoshi, I. Shirawaka, A new algorithm for generating all the independent sets, SIAM Journal on Computing 6 (1977) 505-517]. When G is a bipartite graph, we show that the Galois lattice of G is a partial graph of T(G) and we deduce that algorithms based on the Galois lattice are a particular search of T(G). Moreover, we show that algorithms in [G. Alexe, S. Alexe, Y. Crama, S. Foldes, P.L. Hammer, B. Simeone, Consensus algorithms for the generation of all maximal bicliques, Discrete Applied Mathematics 145 (1) (2004) 11-21; L. Nourine, O. Raynaud, A fast algorithm for building lattices, Information Processing Letters 71 (1999) 199-204] generate maximal bicliques of a bipartite graph in O(n^2) per maximal biclique, where n is the number of vertices in G. Finally, we show that under some specific numbering, the transition graph T(G) has a hamiltonian path for chordal and comparability graphs.

On the Enumeration of Minimal Dominating Sets and Related Notions

A dominating set

A fast incremental algorithm for building lattices

D

Enumeration of minimal dominating sets and variants

in a graph is a subset of its vertex set such that each vertex is either in

A Unified Hierarchy for Functional Dependencies, Conditional Functional Dependencies and Association Rules

D

Protocol-Based Web Service Composition

or has a neighbour in

Uncovering and reducing hidden combinatorics in guigues-duquenne bases

D

Tree structure for distributive lattices and its applications

. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Secondly, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs, and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in

On the Neighbourhood Helly of Some Graph Classes and Applications to the Enumeration of Minimal Dominating Sets

P_6

Extending set-based dualization: application to pattern mining

-free chordal graphs, a proper superclass of split graphs. Finally, we investigate the complexity of the enumeration of (inclusion-wise) minimal connected dominating sets and minimal total dominating sets of graphs. We show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: minimal total dominating sets, minimal total dominating sets in split graphs, minimal connected dominating sets in split graphs, minimal dominating sets in co-bipartite graphs.