In this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. We focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of Fomin and Reading, and colored quiver mutation.

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) \N then exists v ∈ N such that d(u, v) ≤ l). A k-kernel is a (k, k − 1)-kernel. We propose an extension of the definition of (k, l)-kernel to (arc-)weighted digraphs, verif...

Abstract. The shortest paths tree problem consists in finding a spanning tree rooted at a given node, in a directed weighted graph, such that for each node i , the path of the tree which goes from i to the root has minimal weight. We propose an algorithm which is a deterministic version of Howard’s policy iteration scheme. We show that policy iteration is faster than the Bellman (or value itera...

Article history: Available online 10 April 2013

A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph. In the present paper, we propose po...

The present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete-time multi-agent systems. In the systems considered here, both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time. Under the hypothesis that some spanning tree structure is preserved along time, and that some nonzero mi...

This article is devoted to estimating the speed of convergence towards consensus for a general class of discrete-time multi-agent systems. In the systems considered here, both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time. Under the hypothesis that some spanning tree structure is preserved along time, and that some nonzero minimal...