Arnaud Mary

On the Enumeration of Minimal Dominating Sets and Related Notions

A dominating set

Enumeration of minimal dominating sets and variants

D

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

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

On the Enumeration and Counting of Minimal Dominating sets in Interval and Permutation Graphs

D

Polynomial Delay Algorithm for Listing Minimal Edge Dominating Sets in Graphs

or has a neighbour in

A Combinatorial Algorithm for Microbial Consortia Synthetic Design.

D

BacHBerry: BACterial Hosts for production of Bioactive phenolics from bERRY fruits

. 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

Algorithms for k-meet-semidistributive lattices

P_6

Bounding the Order of a Graph Using Its Diameter and Metric Dimension: A Study Through Tree Decompositions and VC Dimension

-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.

An average study of hypergraphs and their minimal transversals

In this paper, we are interested in the enumeration of minimal dominating sets in graphs. A polynomial delay algorithm with polynomial space in split graphs is presented. We then introduce a notion of maximal extension (a set of edges added to the graph) that keeps invariant the set of minimal dominating sets, and show that graphs with extensions as split graphs are exactly the ones having chordal graphs as extensions. We finish by relating the enumeration of some variants of dominating sets to the enumeration of minimal transversals in hypergraphs.