Binh-Minh Bui-Xuan

Binh-Minh Bui-Xuan, Florent Krasnopol, Bruno Monasson et al.

Temporal graphs are graphs where the presence or properties of their vertices and edges change over time. When time is discrete, a temporal graph can be defined as a sequence of static graphs over a discrete time span, called lifetime, or as a single graph where each edge is associated with a specific set of time instants where the edge is alive. For static graphs, Courcelle's Theorem asserts that any graph problem expressible in monadic second-order logic can be solved in linear time on graphs of bounded tree-width. We propose the first adaptation of Courcelle's Theorem for monadic second-order logic on temporal graphs that does not explicitly rely on a parameter proportional to the lifetime, or defined as the maximum number of time-edges incident with any vertex which in the worst case is higher than the lifetime. We then introduce the notion of derivative over a sliding time window of a chosen size, and define the tree-width and twin-width of the temporal graph's derivative. We exemplify its usefulness with meta-theorems with respect to a temporal variant of first-order logic. The resulting logic expresses a wide range of temporal graph problems including a version of temporal cliques, an important notion when querying time series databases for community structures.

Binh-Minh Bui-Xuan, Nhat-Minh Nguyen, Sébastien Tixeuil et al.

Dynamic networks are graphs in which edges are available only at specific time instants, modeling connections that change over time. The dynamic network creation game studies this setting as a strategic interaction where each vertex represents a player. Players can add or remove time-labeled edges in order to minimize their personal cost. This cost has two components: a construction cost, calculated as the number of time instants during which a player maintains edges multiplied by a constant $α$, and a communication cost, defined as the average distance to all other vertices in the network. Communication occurs through temporal paths, which are sequences of adjacent edges with strictly increasing time labels and no repeated vertices. We show for the shortest distance (minimizing the number of edges) that the price of anarchy can be proportional to the number of vertices, contrasting the constant price conjectured for static networks.

Binh-Minh Bui-Xuan, Michel Habib, Fabien de Montgolfier et al.

It is well-known since the seventies of last century that Depth First Search (DFS) can be used to compute strongly connected components [RE. Tarjan. SIAM Journal on Computing, 1972] and Breadth First Search (BFS) can be used to compute distance in graphs [GY. Handler. Transportation Science, 1973]. We furthermore demonstrate that these standard graph searches are powerful enough to recognize and certify several well-structured graph classes. Specifically, we provide a single DFS approach for recognizing and certifying trivially perfect graphs that is significantly simpler than previous methods using [FPM. Chu. Information Processing Letters, 2008]. We further show that a single BFS can recognize split graphs and bipartite chain graphs, and we improve upon the triple LexBFS algorithm for proper interval graphs [DG. Corneil. Discrete Applied Mathematics, 2004] by proposing a two consecutive BFS recognition scheme. These results are underpinned by characterizations using vertex orderings that avoid specific patterns [L. Feuilloley, M. Habib. SIAM Journal on Discrete Mathematics, 2021]. Finally, we provide a structural study of connected proper interval graphs, proving that their characterizations via special orderings are unique up to reversal and the permutation of true twins.