Undirected graph
View article: Blind cop-width and balanced minors of graphs
Blind cop-width and balanced minors of graphs Open
We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices i…
View article: Blind cop-width and balanced minors of graphs
Blind cop-width and balanced minors of graphs Open
We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices i…
View article: Acyclic dichromatic number of oriented graphs
Acyclic dichromatic number of oriented graphs Open
The dichromatic number $\vecχ(D)$ of a digraph $D=(V,A)$ is the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ into $k$ subsets so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$. This is a generalizati…
View article: New Algorithms and Hardness Results for Connected Clustering
New Algorithms and Hardness Results for Connected Clustering Open
Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph $G$ that can be completely unrelated to the metric. The aim is to partition the $n$ verti…
View article: An Approximate Solution to the Minimum Vertex Cover Problem: The Hvala Algorithm
An Approximate Solution to the Minimum Vertex Cover Problem: The Hvala Algorithm Open
The Minimum Vertex Cover (MVC) problem is a fundamental NP-complete problem in graph theory that seeks the smallest set of vertices covering all edges in an undirected graph G = (V, E). This paper presents the find_vertex_cover algorithm, …
View article: Joint learning of a network of linear dynamical systems via total variation penalization
Joint learning of a network of linear dynamical systems via total variation penalization Open
We consider the problem of joint estimation of the parameters of $m$ linear dynamical systems, given access to single realizations of their respective trajectories, each of length $T$. The linear systems are assumed to reside on the nodes …
View article: New Algorithms and Hardness Results for Connected Clustering
New Algorithms and Hardness Results for Connected Clustering Open
Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph $G$ that can be completely unrelated to the metric. The aim is to partition the $n$ verti…
View article: Joint learning of a network of linear dynamical systems via total variation penalization
Joint learning of a network of linear dynamical systems via total variation penalization Open
We consider the problem of joint estimation of the parameters of $m$ linear dynamical systems, given access to single realizations of their respective trajectories, each of length $T$. The linear systems are assumed to reside on the nodes …
View article: Enumerating Acyclic Orientations with Prescribed Source and Sink Vertices: A Recursive Framework
Enumerating Acyclic Orientations with Prescribed Source and Sink Vertices: A Recursive Framework Open
Acyclic orientations are fundamental objects in graph theory and combinatorics, with broad applications ranging from partial order theory and scheduling to network flows and computational complexity. The enumeration of all acyclic orientat…
View article: Approximating maximum properly colored forests via degree bounded independent sets
Approximating maximum properly colored forests via degree bounded independent sets Open
In the Maximum-size Properly Colored Forest problem, we are given an edge-colored undirected graph and the goal is to find a properly colored forest with as many edges as possible. We study this problem within a broader framework by introd…
View article: Enumerating Acyclic Orientations with Prescribed Source and Sink Vertices: A Recursive Framework
Enumerating Acyclic Orientations with Prescribed Source and Sink Vertices: A Recursive Framework Open
Acyclic orientations are fundamental objects in graph theory and combinatorics, with broad applications ranging from partial order theory and scheduling to network flows and computational complexity. The enumeration of all acyclic orientat…
View article: Approximating maximum properly colored forests via degree bounded independent sets
Approximating maximum properly colored forests via degree bounded independent sets Open
In the Maximum-size Properly Colored Forest problem, we are given an edge-colored undirected graph and the goal is to find a properly colored forest with as many edges as possible. We study this problem within a broader framework by introd…
View article: Enumerating Acyclic Orientations with Prescribed Source and Sink Vertices: A Recursive Framework
Enumerating Acyclic Orientations with Prescribed Source and Sink Vertices: A Recursive Framework Open
Acyclic orientations are fundamental objects in graph theory and combinatorics, with broad applications ranging from partial order theory and scheduling to network flows and computational complexity. The enumeration of all acyclic orientat…
View article: Biclosed monoidal structures on the categories of digraphs and graphs
Biclosed monoidal structures on the categories of digraphs and graphs Open
We show that the categories of directed and undirected reflexive graphs carry exactly two (up to isomorphism) biclosed monoidal structures.
View article: Biclosed monoidal structures on the categories of digraphs and graphs
Biclosed monoidal structures on the categories of digraphs and graphs Open
We show that the categories of directed and undirected reflexive graphs carry exactly two (up to isomorphism) biclosed monoidal structures.
View article: Optimal graph joining with applications to isomorphism detection and identification
Optimal graph joining with applications to isomorphism detection and identification Open
We introduce an optimal transport based approach for comparing undirected graphs with non-negative edge weights and general vertex labels, and we study connections between the resulting linear program and the graph isomorphism problem. Our…
View article: Optimal graph joining with applications to isomorphism detection and identification
Optimal graph joining with applications to isomorphism detection and identification Open
We introduce an optimal transport based approach for comparing undirected graphs with non-negative edge weights and general vertex labels, and we study connections between the resulting linear program and the graph isomorphism problem. Our…
View article: An Approximate Solution to the Minimum Vertex Cover Problem: The Hvala Algorithm
An Approximate Solution to the Minimum Vertex Cover Problem: The Hvala Algorithm Open
The Minimum Vertex Cover (MVC) problem is a fundamental NP-complete problem in graph theory that seeks the smallest set of vertices covering all edges in an undirected graph G = (V, E). This paper presents the find_vertex_cover algorithm, …
View article: Directed Hamiltonicity in Generalized Kneser Graphs
Directed Hamiltonicity in Generalized Kneser Graphs Open
We prove that the canonical orientation of the generalized Kneser graph $KG(n,k,s)$ contains a directed Hamiltonian cycle for all integers $s \geq 3$ and $n>sk$. Furthermore, we establish that the dichromatic number of this oriented graph …
View article: Directed Hamiltonicity in Generalized Kneser Graphs
Directed Hamiltonicity in Generalized Kneser Graphs Open
We prove that the canonical orientation of the generalized Kneser graph $KG(n,k,s)$ contains a directed Hamiltonian cycle for all integers $s \geq 3$ and $n>sk$. Furthermore, we establish that the dichromatic number of this oriented graph …
View article: Coloured Gaussian directed acyclic graphical models
Coloured Gaussian directed acyclic graphical models Open
We study submodels of Gaussian directed acyclic graph (DAG) models defined by partial homogeneity constraints imposed on the model error variances and structural coefficients. We represent these models with coloured DAGs and investigate th…
View article: The Structure of Cayley Graph of Dihedral Groups of Valency 4
The Structure of Cayley Graph of Dihedral Groups of Valency 4 Open
Let G be a group and S be a subset of G in which e /∈ S and S−1 ⊆ S. The Cayley graph of group G with respect to subset S, denoted by Cay(G, S), is an undirected simple graph whose vertices are all elements of G, and two vertices x and y a…
View article: Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers
Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers Open
We present the first deterministic nearly-linear time algorithm for single-source shortest paths with negative edge weights on directed graphs: given a directed graph $G$ with $n$ vertices, $m$ edges whose weights are integer in $\{-W,\dot…
View article: Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers
Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers Open
We present the first deterministic nearly-linear time algorithm for single-source shortest paths with negative edge weights on directed graphs: given a directed graph $G$ with $n$ vertices, $m$ edges whose weights are integer in $\{-W,\dot…
View article: Improved Tree Sparsifiers in Near-Linear Time
Improved Tree Sparsifiers in Near-Linear Time Open
A \emph{tree cut-sparsifier} $T$ of quality $α$ of a graph $G$ is a single tree that preserves the capacities of all cuts in the graph up to a factor of $α$. A \emph{tree flow-sparsifier} $T$ of quality $α$ guarantees that every demand tha…
View article: Improved Tree Sparsifiers in Near-Linear Time
Improved Tree Sparsifiers in Near-Linear Time Open
A \emph{tree cut-sparsifier} $T$ of quality $α$ of a graph $G$ is a single tree that preserves the capacities of all cuts in the graph up to a factor of $α$. A \emph{tree flow-sparsifier} $T$ of quality $α$ guarantees that every demand tha…
View article: The Rainbow Arborescence Problem on Cycles
The Rainbow Arborescence Problem on Cycles Open
The rainbow arborescence conjecture posits that if the arcs of a directed graph with $n$ vertices are colored by $n-1$ colors such that each color class forms a spanning arborescence, then there is a spanning arborescence that contains exa…
View article: Sign games on graphs
Sign games on graphs Open
We define the Sign Game as a two-player game played on a simple undirected mathematical graph $G$. The players alternate turns, assigning vertices of $G$ either $1$ or $-1$, and edges take on the value of the product of their endvertices. …
View article: The Rainbow Arborescence Problem on Cycles
The Rainbow Arborescence Problem on Cycles Open
The rainbow arborescence conjecture posits that if the arcs of a directed graph with $n$ vertices are colored by $n-1$ colors such that each color class forms a spanning arborescence, then there is a spanning arborescence that contains exa…
View article: Improved Additive Approximation Algorithms for APSP
Improved Additive Approximation Algorithms for APSP Open
The All-Pairs Shortest Paths (APSP) is a foundational problem in theoretical computer science. Approximating APSP in undirected unweighted graphs has been studied for many years, beginning with the work of Dor, Halperin and Zwick [SICOMP'0…