Pedro Montealegre

The Simultaneous Number-in-Hand Communication Model for Networks: Private Coins, Public Coins and Determinism

We study the multiparty communication model where players are the nodes of a network and each of these players knows his/her own identifier together with the identifiers of his/her neighbors. The players simultaneously send a unique message to a referee who must decide a graph property. The goal of this article is to separate, from the point of view of message size complexity, three different settings: deterministic protocols, randomized protocols with private coins and randomized protocols with public coins. For this purpose we introduce the boolean function Twins. This boolean function returns 1 if and only if there are two nodes with the same neighborhood.

Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw). We prove that for any graph, the number of minimal separators is \(\mathcal{O}^*(3^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.6181^{\operatorname{mw}})\), the number of potential maximal cliques is \(\mathcal{O}^*(4^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\), and these objects can be listed within the same running times. (The \(\mathcal{O}^*\) notation suppresses polynomial factors in the size of the input.) Combined with known results [3,12], we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time \(\mathcal{O}^*(4^{\operatorname{vc}})\) or \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\).

On Distance-d Independent Set and Other Problems in Graphs with few Minimal Separators

Fomin and Villanger [14], STACS 2010 proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let

Fixing improper colorings of graphs

\mathcal {G}_{{\text {poly}}}

A Fast Parallel Algorithm for the Robust Prediction of the Two-Dimensional Strict Majority Automaton

be the class of graphs with at most

Beyond Classes of Graphs with Few Minimal Separators: FPT Results Through Potential Maximal Cliques

{\text {poly}}n