## Section: New Results

### Graph and Combinatorial Algorithms

#### Collision-Free Network Exploration

In the collision-free exploration model considered
in [16] , a set of mobile agents is placed at different
nodes of a $n$-node network. The agents synchronously move along the network
edges in a collision-free way, i.e., in no round may two agents occupy the same
node. In each round, an agent may choose to stay at its currently occupied node
or to move to one of its neighbors. An agent has no knowledge of the number and
initial positions of other agents. We are looking for the shortest possible time
required to complete the collision-free *network exploration*, i.e., to
reach a configuration in which each agent is guaranteed to have visited all
network nodes and has returned to its starting location.

In this work, we first considered the scenario when each mobile agent knows the map of the network, as well as its own initial position. We established a connection between the number of rounds required for collision-free exploration and the degree of the minimum-degree spanning tree of the graph. We provided tight (up to a constant factor) lower and upper bounds on the collision-free exploration time in general graphs, and the exact value of this parameter for trees. For our second scenario, in which the network is unknown to the agents, we proposed collision-free exploration strategies running in $O\left({n}^{2}\right)$ rounds for tree networks and in $O({n}^{5}logn)$ rounds for general networks.

#### Properties of Graph Search Procedures

In [4] , we study the last vertex discovered by a graph search such as BFS or DFS. End-vertices of a given graph search may have some nice properties (as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial). Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break rule can determine the complexity of the end-vertex problem for BFS or DFS. In particular we prove a counter-intuitive NP-completeness result for Breadth First Search, answering a question of D.G. Corneil, E. Köhler and J-M Lanlignel.

#### Matchings in Hypergraphs

A rainbow matching for (not necessarily distinct) sets ${F}_{1},...{F}_{k}$ of hypergraph edges is a matching consisting of $k$ edges, one from each ${F}_{i}$. The aim of [3] is twofold—to put order in the multitude of conjectures that relate to this concept (some first presented here), and to prove partial results on one of the central conjectures settled by Ryser, Brualdi and Stein.

#### Common Intervals and Application to Genome Comparison

In [6] , we show how to identify generalized common and conserved nested intervals. This is a bio-informatics papers, explaining how to compute more relaxed variants of common or of conserved intervals of two permutations, which has applications in genome comparison. It also presents some properties of the family of intervals, useful for storing them.

#### Graph Decomposition

In [10] , we present a general framework for computing a large family of graph decomposition, the H-join. It generalizes some well know tools like modular decomposition or split decomposition. The paper explains how to compute it in polynomial time. A new canonical decomposition for sesquiprime graphs is also presented.

#### Combinatorial Optimization

Normal cone and subdifferential have been generalized through various continuous functions; in [8] , we focus on a non separable $Q$-subdifferential version. Necessary and sufficient optimality conditions for unconstrained nonconvex problems are revisited accordingly. For inequality constrained problems, $Q$-subdifferential and the lagrangian multipliers, enhanced as continuous functions instead of scalars, allow us to derive new necessary and sufficient optimality conditions. In the same way, the Legendre-Fenchel conjugate is generalized into $Q$-conjugate and global optimality conditions are derived by $Q$-conjugate as well, leading to a tighter inequality.