#### 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

Complexity and equivalency of multiset dimension and ID-colorings

Dipayan Chakraborty, Florent Foucaud, Soumen Nandi, Sagnik Sen, D K Supraja - Feb. 9, 2023

New Bounds and Constructions for Neighbor-Locating Colorings of Graphs

9th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM 2023)

Florent Foucaud, Krishna Narayanan, Lekshmi Ramasubramony Sulochana - Feb. 9, 2023

Monitoring edge-geodetic sets in graphs

9th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM 2023)

Dipayan Chakraborty, Florent Foucaud, Aline Parreau, Annegret Wagler - Feb. 9, 2023

On Three Domination-Based Identification Problems in Block Graphs

9th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM 2023)

Sandip Das, Florent Foucaud, Sk Samim Islam, Joydeep Mukherjee - Feb. 9, 2023

Relation Between Broadcast Domination and Multipacking Numbers on Chordal Graphs

9th International Conference on Algorithms and Discrete Applied Mathematics (CALDAM 2023)

Antoine Dailly, Elżbieta Sidorowicz - Jan. 16, 2023

Neighbour sum distinguishing edge-weightings with local constraints

Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca - Dec. 19, 2022

On graphs coverable by k shortest paths

33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Dibyayan Chakraborty, Antoine Dailly, Sandip Das, Florent Foucaud, Harmender Gahlawat, Subir Kumar Ghosh - Dec. 19, 2022

Complexity and Algorithms for ISOMETRIC PATH COVER on Chordal Graphs and Beyond

33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Florent Foucaud, Tuomo Lehtilä - Dec. 1, 2022

Revisiting and Improving Upper Bounds for Identifying Codes

SIAM Journal on Discrete Mathematics

Dipayan Chakraborty, Florent Foucaud, Anni Hakanen, Michael A. Henning, Annegret K. Wagler - Nov. 28, 2022

Progress towards the two-thirds conjecture on locating-total dominating sets

All publications are here