Related papers: On Mimicking Networks Representing Minimum Termina…

Given a graph $G$ and a set of terminals $T$, a \emph{distance emulator} of $G$ is another graph $H$ (not necessarily a subgraph of $G$) containing $T$, such that all the pairwise distances in $G$ between vertices of $T$ are preserved in…

We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor the vertices and/or the edges of a network (viewed as a graph) in order to detect and prevent failures. Inspired by two…

We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ of vertices of a graph $G$ is a \emph{distance-edge-monitoring set} if for every edge $e$ of $G$, there is a vertex $x$ of $M$ and a vertex $y$ of $G$…

We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T \subseteq V(G)$, it asks whether there is a vertex set $S \subseteq V(G) \setminus T$ of size at most…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…

We study the following characterization problem. Given a set $T$ of terminals and a $(2^{|T|}-2)$-dimensional vector $\pi$ whose coordinates are indexed by proper subsets of $T$, is there a graph $G$ that contains $T$, such that for all…

Let $G$ be a graph of order $n$ and size $m$, and let $\mathrm{mc}_{k}\left( G\right) $ be the maximum size of a $k$-cut of $G.$ It is shown that \[ \mathrm{mc}_{k}\left( G\right) \leq\frac{k-1}{k}\left( m-\frac{\mu_{\min }\left( G\right)…

Let $G$ be a nontrivial connected graph of order $n$ and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such…

Given a graph $G = (V,E)$ and a terminal $s\in V$, a cut $X$ for $s$ is a vertex set that contains $s$. We look for a cut that is small in two senses, i.e., there are no more than $k$ vertices in $X$ and no more than $t$ edges leaving $X$.…

In 1985, Chv\'{a}tal introduced the concept of star cutsets as a means to investigate the properties of perfect graphs, which inspired many researchers to study cutsets with some specific structures, for example, star cutsets, clique…

Due to the increasing discovery and implementation of networks within all disciplines of life, the study of subgraph connectivity has become increasingly important. Motivated by the idea of community (or sub-graph) detection within a…

Given a graph $G = (V,E)$, a set $T$ of vertex pairs, and an integer $k$, Hitting Geodesic Intervals asks whether there is a set $S \subseteq V$ of size at most $k$ such that for each terminal pair $\{u,v\} \in T$, the set $S$ intersects at…

We introduce a variant of the multiway cut that we call the min-max connected multiway cut. Given a graph $G=(V,E)$ and a set $\Gamma\subseteq V$ of $t$ terminals, partition $V$ into $t$ parts such that each part is connected and contains…

Foucaud et al. [Discrete Appl. Math. 319 (2022), 424-438] recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. For a set $M$ of vertices and an edge $e$ of a graph $G$, let $P(M,…

In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if there is any) separating $S$ from $T$ under the following ``uncut'' condition. In the…

Given a weighted graph $G=(V,E,w)$ with a set of $k$ terminals $T\subset V$, the Steiner Point Removal problem seeks for a minor of the graph with vertex set $T$, such that the distance between every pair of terminals is preserved within a…

The vertex connectivity of a graph $G$ is the size of the smallest set of vertices $S$ such that $G \setminus S$ is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and…

Dumas, Foucaud, Perez, and Todinca [SIAM J. Disc. Math., 2024] proved that if the vertex set of a graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is bounded by $\mathcal{O}(k \cdot 3^k)$. We prove a coarse variant…

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut…

Given an undirected graph $G=(V,E)$ with edge capacities $c_e\geq 1$ for $e\in E$ and a subset $T$ of $k$ vertices called terminals, we say that a graph $H$ is a quality-$q$ cut sparsifier for $G$ iff $T\subseteq V(H)$, and for any…