arXiv subjects

Cédric Bentz

Publications and source records attributed to Cédric Bentz.

10 recordsLinked to original sources

arXiv2026

Antimagicness of join graphs

An antimagic labelling of a graph G = (V,E) is a bijection from E to {1,2,...,|E|}, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is antimagic if it has an antimagic labelling. Hartsfield and Ringel in 1990 stated the celebrated conjecture: Every connected graph other than K_2 is antimagic. We prove the conjecture for join graphs with at least three vertices.

math.CO↗

arXiv2026

Antimagic labelling of graphs with maximum degree $Δ(G) = n - 4$

An antimagic labelling of a graph $G = (V,E)$ is a bijection from $E$ to $\{1,2, \ldots, |E|\}$, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is said to be antimagic if it has an antimagic labelling. It has been proven that graphs $G$ with $Δ(G) \geq n - 3$ are antimagic, where $Δ(G)$ is the maximum degree of a vertex in $G$ and $n = |V|$. In this article, we extend this result to graphs with $Δ(G) = n - 4$, provided that $|E| \geq 7n$.

math.CO↗

arXiv2025

Antimagicness of graphs with a dominating clique

A graph $G = (V, E)$ is called antimagic if there exists a bijective labelling $f : E \rightarrow \{1, 2, \ldots, |E|\}$ such that the vertex-sums of labels over edges incident to a given vertex are all distinct. In this paper, we extend the antimagicness results over graphs with a dominating clique. We also introduce an alternative to the usual definition of antimagic graphs, called C-antimagic, allowing for the labelling to be injective in $\{1, 2, . . . , |E| + C\}$ instead of bijective, and show that almost all graphs with a dominating clique are 3-antimagic.

math.CO↗

arXiv2025

Antimagic labellings of (k, 2)-bipartite biregular graphs

An antimagic labelling of a graph is a bijection from the set of edges to $\{1, 2, \ldots , m\}$, such that all vertex-sums are pairwise distinct, where the vertex-sum of a vertex is the sum of labels on the edges incident to it. We say a graph is antimagic if it has an antimagic labelling. In 2023, it has been proven that connected $(k, l)$-bipartite graphs are antimagic if $k \geq l + 2$ and one of k or l is odd. In this paper, we extend this result to connected $(k, 2)$-bipartite biregular graphs for $k \geq 4$ even, and to $(k, 2)$-bipartite biregular graphs for $k \geq 3$ odd.

math.CO↗

arXiv2018

Formulations for designing robust networks. An application to wind power collection

We are interested in the design of survivable capacitated rooted Steiner networks. Given a graph G = (V, E), capacity and cost functions on E, a root r, a subset T of V of terminals and an integer k, we search for a minimum cost subset E $\subset$ E, covering T and r, such that the network induced by E is k-survivable: after the removal of any k edges, there still exists a feasible flow from r to T. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set, a flow and a bi-level formulation where the second-level is a min-max problem (with an attacker and a defender). We propose algorithms for each problem formulation and compare their efficiency.

cs.DS↗

arXiv2018

Robust capacitated trees and networks with uniform demands

We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in case of terminals with uniform demands. Formally, we are given a graph, capacity and cost functions on the edges, a root, a subset of nodes called terminals, and a bound k on the number of edge failures. We first study the problem where k = 1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Second, we consider the problem of computing a minimum-cost survivable network, i.e., a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set based formulation, a flow based one, and a bilevel one (with an attacker and a defender). We propose algorithms to solve each formulation and compare their efficiency.

math.OC↗

arXiv2017

Weighted and locally bounded list-colorings in split graphs, cographs, and partial k-trees

For a fixed number of colors, we show that, in node-weighted split graphs, cographs, and graphs of bounded tree-width, one can determine in polynomial time whether a proper list-coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists. We also show that this result is tight in some sense, by providing hardness results for the cases where any one of the assumptions does not hold. The edge-coloring variant is also studied, as well as special cases of cographs and split graphs.

cs.DM↗

arXiv2017

An FPT algorithm for planar multicuts with sources and sinks on the outer face

Given a list of k source-sink pairs in an edge-weighted graph G, the minimum multicut problem consists in selecting a set of edges of minimum total weight in G, such that removing these edges leaves no path from each source to its corresponding sink. To the best of our knowledge, no non-trivial FPT result for special cases of this problem, which is APX-hard in general graphs for any fixed k>2, is known with respect to k only. When the graph G is planar, this problem is known to be polynomial-time solvable if k=O(1), but cannot be FPT with respect to k under the Exponential Time Hypothesis. In this paper, we show that, if G is planar and in addition all sources and sinks lie on the outer face, then this problem does admit an FPT algorithm when parameterized by k (although it remains APX-hard when k is part of the input, even in stars). To do this, we provide a new characterization of optimal solutions in this case, and then use it to design a "divide-and-conquer" approach: namely, some edges that are part of any such solution actually define an optimal solution for a polynomial-time solvable multiterminal variant of the problem on some of the sources and sinks (which can be identified thanks to a reduced enumeration phase). Removing these edges from the graph cuts it into several smaller instances, which can then be solved recursively.

cs.DS↗

arXiv2016

A note on "Approximation schemes for a subclass of subset selection problems", and a faster FPTAS for the Minimum Knapsack Problem

Pruhs and Woeginger prove the existence of FPTAS's for a general class of minimization and maximization subset selection problems. Without losing generality from the original framework, we prove how better asymptotic worst-case running times can be achieved if a $ρ$-approximation algorithm is available, and in particular we obtain matching running times between maximization and minimization subset selection problems. We directly apply this result to the Minimum Knapsack Problem, for which the original framework yields an FPTAS with running time $O(n^5/ε)$, where $ε$ is the required accuracy and $n$ is the number of items, and obtain an FPTAS with running time $O(n^3/ε)$, thus improving the running time by a quadratic factor in the worst case.

cs.CC↗

arXiv2012

A polynomial-time algorithm for planar multicuts with few source-sink pairs

Given an edge-weighted undirected graph and a list of k source-sink pairs of vertices, the well-known minimum multicut problem consists in selecting a minimum-weight set of edges whose removal leaves no path between every source and its corresponding sink. We give the first polynomial-time algorithm to solve this problem in planar graphs, when k is fixed. Previously, this problem was known to remain NP-hard in general graphs with fixed k, and in trees with arbitrary k; the most noticeable tractable case known so far was in planar graphs with fixed k and sources and sinks lying on the outer face.

cs.DM↗