arXiv subjects

Binh-Minh Bui-Xuan

Publications and source records attributed to Binh-Minh Bui-Xuan.

11 recordsLinked to original sources

arXiv2026

Model checking with temporal graphs and their derivative

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.

cs.DS↗

arXiv2026

On the power of standard DFS and BFS

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.

cs.DS↗

arXiv2026

The Price of Universal Temporal Reachability

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.

cs.DS↗

arXiv2024

Distributed computation of temporal twins in periodic undirected time-varying graphs

Twin nodes in a static network capture the idea of being substitutes for each other for maintaining paths of the same length anywhere in the network. In dynamic networks, we model twin nodes over a time-bounded interval, noted $(Δ,d)$-twins, as follows. A periodic undirected time-varying graph $\mathcal G=(G_t)_{t\in\mathbb N}$ of period $p$ is an infinite sequence of static graphs where $G_t=G_{t+p}$ for every $t\in\mathbb N$. For $Δ$ and $d$ two integers, two distinct nodes $u$ and $v$ in $\mathcal G$ are $(Δ,d)$-twins if, starting at some instant, the outside neighbourhoods of $u$ and $v$ has non-empty intersection and differ by at most $d$ elements for $Δ$ consecutive instants. In particular when $d=0$, $u$ and $v$ can act during the $Δ$ instants as substitutes for each other in order to maintain journeys of the same length in time-varying graph $\mathcal G$. We propose a distributed deterministic algorithm enabling each node to enumerate its $(Δ,d)$-twins in $2p$ rounds, using messages of size $O(δ_\mathcal G\log n)$, where $n$ is the total number of nodes and $δ_\mathcal G$ is the maximum degree of the graphs $G_t$'s. Moreover, using randomized techniques borrowed from distributed hash function sampling, we reduce the message size down to $O(\log n)$ w.h.p.

cs.DS↗

arXiv2020

Intersection graphs of almost subnormal subgroups in general skew linear groups

Let $D$ be a division ring, $n$ a positive integer, and GL$_n(D)$ the general linear group of degree $n$ over $D$. In this paper, we study the induced subgraph of the intersection graph of GL$_n(D)$ generated by all non-trivial proper almost subnormal subgroups of GL$_n(D)$. We show that this subgraph is complete if it is non-null. This property will be used to study subgroup structure of a division ring. In particular, we prove that every non-central almost subnormal subgroup of the multiplicative group $D^*$ of a division ring $D$ contains a non-central subnormal subgroup of $D^*$.

math.RA↗

arXiv2019

Lightweight FEC: Rectangular Codes with Minimum Feedback Information

We propose a hybrid protocol combining a rectangular error-correcting code - paired with an error-detecting code - and a backward error correction in order to send packages of information over a noisy channel. We depict a linear-time algorithm the receiver can use to determine the minimum amount of information to be requested from the sender in order to repair all transmission errors. Repairs may possibly occur over several cycles of emissions and requests. We show that the expected bandwidth use on the backward channel by our protocol is asymptotically small. In most configurations we give the explicit asymptotic expansion for said expectation. This is obtained by linking our problem to a well known algorithmic problem on a gadget graph, feedback edge set. The little use of the backward channel makes our protocol suitable where one could otherwise simply use backward error correction, e.g. TCP, but where overly using the backward channel is undesirable. We confront our protocol to numerical analysis versus TCP protocol. In most cases our protocol allows to reduce the number of iterations down to 60%, while requiring only negligibly more packages.

cs.NI↗

arXiv2019

Temporal Matching

A link stream is a sequence of pairs of the form $(t,\{u,v\})$, where $t\in\mathbb N$ represents a time instant and $u\neq v$. Given an integer $γ$, the $γ$-edge between vertices $u$ and $v$, starting at time $t$, is the set of temporally consecutive edges defined by $\{(t',\{u,v\}) | t' \in [t,t+γ-1]\}$. We introduce the notion of temporal matching of a link stream to be an independent $γ$-edge set belonging to the link stream. We show that the problem of computing a temporal matching of maximum size is NP-hard as soon as $γ>1$. We depict a kernelization algorithm parameterized by the solution size for the problem. As a byproduct we also give a $2$-approximation algorithm. Both our $2$-approximation and kernelization algorithms are implemented and confronted to link streams collected from real world graph data. We observe that finding temporal matchings is a sensitive question when mining our data from such a perspective as: managing peer-working when any pair of peers $X$ and $Y$ are to collaborate over a period of one month, at an average rate of at least two email exchanges every week. We furthermore design a link stream generating process by mimicking the behaviour of a random moving group of particles under natural simulation, and confront our algorithms to these generated instances of link streams. All the implementations are open source.

cs.DS↗

arXiv2014

A Note on Graphs of Linear Rank-Width 1

We prove that a connected graph has linear rank-width 1 if and only if it is a distance-hereditary graph and its split decomposition tree is a path. An immediate consequence is that one can decide in linear time whether a graph has linear rank-width at most 1, and give an obstruction if not. Other immediate consequences are several characterisations of graphs of linear rank-width 1. In particular a connected graph has linear rank-width 1 if and only if it is locally equivalent to a caterpillar if and only if it is a vertex-minor of a path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors of graphs of small tree-width, arxiv:1203.3606] if and only if it does not contain the co-K_2 graph, the Net graph and the 5-cycle graph as vertex-minors [Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for linear rank-width at most 1, arxiv:1106.2533].

cs.DM↗

arXiv2013

How modular structure can simplify tasks on networks

By considering the task of finding the shortest walk through a network we find an algorithm for which the run time is not as O(2^n), with n being the number of nodes, but instead scales with the number of nodes in a coarsened network. This coarsened network has a number of nodes related to the number of dense regions in the original graph. Since we exploit a form of local community detection as a preprocessing, this work gives support to the project of developing heuristic algorithms for detecting dense regions in networks: preprocessing of this kind can accelerate optimization tasks on networks. Our work also suggests a class of empirical conjectures for how structural features of efficient networked systems might scale with system size.

cs.SI↗

arXiv2007

Algorithmic Aspects of a General Modular Decomposition Theory

A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs, the terminology ``module'' not only captures the classical graph modules but also allows to handle 2-connected components, star-cutsets, and other vertex subsets. The main result is that most of the nice algorithmic tools developed for modular decomposition of graphs still apply efficiently on our generalisation of modules. Besides, when an essential axiom is satisfied, almost all the important properties can be retrieved. For this case, an algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is generalised and yields a very efficient solution to the associated decomposition problem.

cs.DS↗

arXiv2007

Unifying two Graph Decompositions with Modular Decomposition

We introduces the umodules, a generalisation of the notion of graph module. The theory we develop captures among others undirected graphs, tournaments, digraphs, and $2-$structures. We show that, under some axioms, a unique decomposition tree exists for umodules. Polynomial-time algorithms are provided for: non-trivial umodule test, maximal umodule computation, and decomposition tree computation when the tree exists. Our results unify many known decomposition like modular and bi-join decomposition of graphs, and a new decomposition of tournaments.

cs.DS↗