A temporal graph is a graph for which the edge set can change from one time step to the next. This paper considers undirected temporal graphs defined over L time steps and connected at each time step. We study the Shortest Temporal Exploration Problem (STEXP) that, given all the evolution of the graph, asks for a temporal walk that starts at a given vertex, moves over at most one edge at each time step, visits all the vertices, takes at most L time steps and traverses the smallest number of edges. . We prove that every constantly connected temporal graph with n vertices can be explored with O(n 1.5 ) edges traversed within O(n 3.5 ) time steps. This result improves the upper bound of O(n 2 ) edges for an exploration provided by the upper bound of time steps for an exploration which is also O(n 2 ). Morever, we study the case where the graph has a diameter bounded by a parameter k at each time step and we prove that there exists an exploration which takes O(kn 2 ) time steps and traverses O(kn) edges. Finally, the case where the underlying graph is a cycle is studied and tight bounds are provided on the number of edges traversed in the worst-case if L $\ge$ 2n -3.

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In this paper we introduce the notion of protein interaction network. This is a graph whose vertices are the protein's amino acids and whose edges are the interactions between them. Using a graph theory approach, we identify a number of properties of these networks. We compare them to the general scale-free network model and we analyze the existence of nodes with a high interaction level.

This paper presents a new network-flow formulation for the problem of predicting 3D protein structures using threading. Several integer-programming models based on this formulation are proposed and compared. These models allow for an efficient decomposition and for the application of a parallel branch-and-cut algorithm, significantly reducing the running time. The efficiency of our approach has been confirmed by extensive computational experiments.