Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv admin note: text overlap with arXiv:1409.060

Sequential Metric Dimension

International audienceSeager introduced the following game in 2013. An invisible and immobile target is hidden at some vertex of a graph $G$. Every step, one vertex $v$ of $G$ can be probed which results in the knowledge of the distance between $v$ and the target. The objective of the game is to minimize the number of steps needed to locate the target, wherever it is. We address the generalization of this game where $k ≥ 1$ vertices can be probed at every step. Our game also generalizes the notion of the metric dimension of a graph. Precisely, given a graph $G$ and two integers $k, ≥ 1$, the Localization Problem asks whether there exists a strategy to locate a target hidden in $G$ in at most steps by probing at most $k$ vertices per step. We show this problem is NP-complete when $k$ (resp.,) is a fixed parameter. Our main results are for the class of trees where we prove this problem is NP-complete when $k$ and are part of the input but, despite this, we design a polynomial-time (+1)-approximation algorithm in trees which gives a solution using at most one more step than the optimal one. It follows that the Localization Problem is polynomial-time solvable in trees if $k$ is fixed

Study of Identifying Code Polyhedra for Some Families of Split Graphs

International audienceThe identifying code problem is a newly emerging search problem, challenging both from a theoretical and a computational point of view, even for special graphs like bipartite graphs and split graphs. Hence, a typical line of attack for this problem is to determine minimum identifying codes of special graphs or to provide bounds for their size. In this work we study the associated polyhedra for some families of split graphs: headless spiders and complete suns. We provide the according linear relaxations, discuss their combinatorial structure, and demonstrate how the associated polyhedra can be entirely described or polyhedral arguments can be applied to find minimum identifying codes for special split graphs. We discuss further lines of research in order to apply similar techniques to obtain strong lower bounds stemming from linear relaxations of the identifying code polyhedron, enhanced by suitable cutting planes to be used in a B&C framework

Polyhedra Associated with Open Locating-Dominating and Locating Total-Dominating Sets in Graphs

International audienceThe problems of determining open locating-dominating or locating total-dominating sets of minimum cardinality in a graph G are variations of the classical minimum dominating set problem in G and are all known to be hard for general graphs. A typical line of attack is therefore to determine the cardinality of minimum such sets in special graphs. In this work we study the two problems from a polyhedral point of view. We provide the according linear relaxations, discuss their combinatorial structure, and demonstrate how the associated polyhedra can be entirely described or polyhedral arguments can be applied to find minimum such sets for special graphs