Related papers: On Conflict-Free Cuts: Algorithms and Complexity

Cutwidth is a widely studied parameter that quantifies how well a graph can be decomposed along small edge-cuts. It complements pathwidth, which captures decomposition by small vertex separators, and it is well-known that cutwidth…

This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. [COCOON 2024]. A path in a vertex-colored graph is called conflict-free if there is a color that appears…

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of…

For a subset $X$ of the vertex set $\VV(\GG)$ of a graph $\GG$, we denote the set of edges of $\GG$ which have exactly one end in $X$ by $\partial(X)$ and refer to it as the cut of $X$ or edge cut $\partial(X)$. A graph $\GG=(\VV,\EE)$ is…

A resolving set for a simple graph $G$ is a subset of vertex set of $G$ such that it distinguishes all vertices of $G$ using the shortest distance from this subset. This subset is a metric basis if it is the smallest set with this property.…

We consider the classical Minimum Balanced Cut problem: given a graph $G$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em…

We study how to sparsify connectivity in graphs under a tight deletion budget. Given a graph $G$ and integers $k,x \ge 0$, Critical Node Cut (CNC) asks whether we can delete at most $k$ vertices so that the number of remaining unordered…

Given a positive integer $d$, the d-CUT is the problem of deciding if an undirected graph $G=(V,E)$ has a cut $(A,B)$ such that every vertex in $A$ (resp. $B$) has at most $d$ neighbors in $B$ (resp. $A$). For $d=1$, the problem is referred…

Let $G$ be a graph with $m$ edges and let $\mathrm{mc}(G)$ denote the size of a largest cut of $G$. The difference $\mathrm{mc}(G)-m/2$ is called the surplus $\mathrm{sp}(G)$ of $G$. A fundamental problem in MaxCut is to determine…

This paper investigates an extremely classic NP-complete problem: How to determine if a graph G, where each vertex has a degree of at most 4, can be 3-colorable(The research in this paper focuses on graphs G that satisfy the condition where…

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, \mrbm: Given an edge-colored graph $G$, how large is the largest rainbow matching in $G$?…

A conflict-free $k$-coloring of a graph $G=(V,E)$ assigns one of $k$ different colors to some of the vertices such that, for every vertex $v$, there is a color that is assigned to exactly one vertex among $v$ and $v$'s neighbors. Such…

We study separating matchings in graphs, that is, matchings whose removal increases the number of connected components, and focus on determining the maximum size of such a matching in a graph $G$, denoted by $\mathrm{mms}(G)$. We show that…

We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum…

Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H} $ and the conflicts to…

The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u,w} if the…

Given an undirected graph $G$, a collection $\{(s_1,t_1),..., (s_k,t_k)\}$ of pairs of vertices, and an integer $p$, the Edge Multicut problem ask if there is a set $S$ of at most $p$ edges such that the removal of $S$ disconnects every…

A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider…

The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph $G$ and demand graph $H$ on a set $T\subseteq V(G)$ of terminals, the task is to find a minimum-weight set $C$ of edges of $G$ such…

We pursue a study of the Generalized Demand Matching problem, a common generalization of the $b$-Matching and Knapsack problems. Here, we are given a graph with vertex capacities, edge profits, and asymmetric demands on the edges. The goal…