#### Theme Algorithmics, Graphs, Complexity

#### Presentation

The main research areas of the team can be classified into two parts : (1) structural graph/lattice theory and the consequences in algorithmics, and (2) algorithmic geometry with a main focus on discrete convexity.

It is usual when studying discrete objects to follow the advice of Descartes, namely **divide and conquer**, *i.e.*, define operators on discrete objects and decompose with respect to these operators in order to explain their inherent complexity. The team has a strong expertise in studying structural graph/lattice theory through the lens of graph/lattice decompositions and their applications in several algorithmic questions and in combinatorics. Some examples of graph decompositions studied in this group are tree-decomposition, rank-decomposition and understanding when the associated width measures are bounded in some graph classes, and also decompositions based on cut set such as clique and stable-cut set and skew partitions with applications in colouring graphs. Some examples of decompositions in lattices are recursive constructions based on copying convex sets or based on colored posets. Some other applications of the studied decompositions can be:

- Listing problems in (hyper)graphs and lattices. We have a strong expertise in the listing of vertex subsets of a hypergraph satisfying a property. We have, for instance, and proved that the listing of minimal dominating sets is equivalent to the well-known Hypergraph Dualisation problem.
- Representation of lattices and efficient algorithms on lattices. This question is mainly studied with respect to the Lattice Dualisation problem, a particular case of which is the Hypergraph Dualisation Problem.
- Optimisation problems in (hyper)graphs. Studied problems include, but are not limited to, colouring problems, graph identification problems, homomorphism questions, etc. We are mostly interested in obtaining either tight bounds on some parameters such as chromatic number or identifying codes, or tight time complexities parametrised by complexity measures.
- Packing and covering problems such as Erdös-Posa functions.

The main interests in algorithmic geometry deal with computational aspects in digital/discrete geometry such as the computation of convex hulls in digitall euclidean spaces. The members are experts in designing heuristics for obtaining competetive algorithms for computing (approximate) convex hulls for various configurations of points in the 2D euclidean space (many of such problems are NP-hard ones). Members often participate to the CG:SHOP challenge, organised during SoCG conference, and win the 2021 edition which were about the coordinated robot motion planning.

Keywords: graph theory; graph complexity parameters; graph decompositions; algorithmic lattice theory; algorithmic geometry; discrete convexity; listing/enumerating algorithms;

### Last publications

Revisiting and Improving Upper Bounds for Identifying Codes

SIAM Journal on Discrete Mathematics

Subhadeep Ranjan Dev, Sanjana Dey, Florent Foucaud, Ralf Klasing, Tuomo Lehtilä - Nov. 15, 2022

The RED-BLUE SEPARATION problem on graphs

Dibyayan Chakraborty, Antoine Dailly, Sandip Das, Florent Foucaud, Harmender Gahlawat, Subir Kumar Ghosh - Oct. 27, 2022

Complexity and algorithms for Isometric Path Cover on chordal graphs and beyond

Fabien Feschet, Jacques-Olivier Lachaud - Oct. 20, 2022

Full Convexity for Polyhedral Models in Digital Spaces

Alexandre Bazin, Laurent Beaudou, Giacomo Kahn, Kaveh Khoshkhah - Oct. 14, 2022

Bounding the Number of Minimal Transversals in Tripartite 3-Uniform Hypergraphs

Florent Foucaud, N. Krishna, Lekshmi R S - Oct. 12, 2022

Monitoring edge-geodetic sets in graphs

Dipayan Chakraborty, Florent Foucaud, Aline Parreau, Annegret K Wagler - Oct. 11, 2022

On three domination-based identification problems in block graphs

Anni Hakanen, Ville Junnila, Tero Laihonen, Ismael G. Yero - Sept. 28, 2022

On the unicyclic graphs having vertices that belong to all their (strong) metric bases

Subhadeep Ranjan Dev, Sanjana Dey, Florent Foucaud, Ralf Klasing, Tuomo Lehtilä - June 7, 2022

The Red-Blue Separation Problem on Graphs

IWOCA 2022

Mamadou Moustapha Kanté, Christophe Paul, Dimitrios M. Thilikos - March 1, 2022

A Linear Fixed Parameter Tractable Algorithm for Connected Pathwidth

SIAM Journal on Discrete Mathematics

All publications are here