Related papers: The metric Menger problem
We introduce an NP-complete graph decision problem, the "Multi-stage graph Simple Path" (abbr. MSP) problem, which focuses on determining the existence of specific "global paths" in a graph $G$. We show that the MSP problem can be solved in…
Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not…
A graph $G=(V,E)$ with geodesic distance $d(\cdot,\cdot)$ is said to be resolved by a non-empty subset $R$ of its vertices when, for all vertices $u$ and $v$, if $d(u,r)=d(v,r)$ for each $r\in R$, then $u=v$. The metric dimension of $G$ is…
We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage…
A (directed) temporal graph is a (directed) graph whose edges are available only at specific times during its (discretized) lifetime $\tau$. In this setting, we ask that walks respect the temporal aspect by defining $\textit{temporal…
We prove that every digraph of circumference $l$ has DAG-width at most $l$ and this is best possible. As a consequence of our result we deduce that the $k$-linkage problem is polynomially solvable for every fixed $k$ in the class of…
In the Min-Sum-Radii (MSR) clustering problem, we are given a finite set X of n points in a metric space. The objective is to find at most k clusters centered at a subset of these points such that every point of X is assigned to one of the…
We study the minimum \emph{Monitoring Edge Geodetic Set} (\megset) problem introduced in [Foucaud et al., CALDAM'23]: given a graph $G$, we say that an edge is monitored by a pair $u,v$ of vertices if \emph{all} shortest paths between $u$…
Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from (Geneson,…
We consider the problem of finding a subgraph of a given graph which minimizes the sum of given functions at vertices evaluated at their subgraph degrees. While the problem is NP-hard already when all functions are the same, we show that it…
Consider the following problem: given a graph with edge costs and a subset Q of vertices, find a minimum-cost subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient…
In this paper, we address the approximate minimization problem of Markov Chains (MCs) from a behavioral metric-based perspective. Specifically, given a finite MC and a positive integer k, we are looking for an MC with at most k states…
Some classical graph problems such as finding minimal spanning tree, shortest path or maximal flow can be done efficiently. We describe slight variations of such problems which are shown to be NP-complete. Our proofs use straightforward…
Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph $G$ of $n$…
Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n,k) is the graph with vertex set $\binom{[n]}{k}$, such that two vertices are adjacent if they are disjoint. We…
Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a…
This work investigates the parameterized complexity of three related graph modification problems. Given a directed graph, a distinguished vertex, and a positive integer k, Minimum Indegree Deletion asks for a vertex subset of size at most k…
The NP-complete $k$-Path problem asks whether a given undirected graph has a (simple) path of length at least $k$. We prove that $k$-Path has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree,…
In a graph, a (perfect) matching cut is an edge cut that is a (perfect) matching. Matching Cut (MC), respectively, Perfect Matching Cut (PMC), is the problem of deciding whether a given graph has a matching cut, respectively, a perfect…
The maximum graph bisection problem is a well known graph partition problem. The problem has been proven to be NP-hard. In the maximum graph bisection problem it is required that the set of vertices is divided into two partition with equal…