Tootfinder

Search-Space Reduction Via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion
Bart M. P. Jansen, Ruben F. A. Verhaegh
https://arxiv.org/abs/2404.09769

Search-Space Reduction Via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion
For an optimization problem $Π$ on graphs whose solutions are vertex sets, a vertex $v$ is called $c$-essential for $Π$ if all solutions of size at most $c \cdot OPT$ contain $v$. Recent work showed that polynomial-time algorithms to detect $c$-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size $k$ of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For …

Trump’s Truth Social shares fall again.
And again, and again, and oops rock bottom.

https://www.axios.com/2024/04/15/trump-truth-social-stock-shares-price-down-why

The stack of $G$-zips is a Mori dream space
Jean-Stefan Koskivirta
https://arxiv.org/abs/2402.09852 https://arxiv.org/pdf/2402.09852<…

The stack of $G$-zips is a Mori dream space
We first extend previous results of the author with T. Wedhorn and W. Goldring regarding the existence of $μ$-ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of $G$-zips of Pink--Wedhorn--Ziegler. Then, we define and study the Cox ring of the stack of $G$-zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of v…

Stability conditions on crepant resolutions of quotients of product varieties
Alexander Perry, Saket Shah
https://arxiv.org/abs/2404.09121 https://<…

Stability conditions on crepant resolutions of quotients of product varieties
We construct stability conditions on crepant resolutions of quotients of certain product varieties, giving as a special case the first examples of stability conditions on strict Calabi-Yau varieties of arbitrary dimension. Along the way, we prove the crepant resolutions are derived equivalent to the corresponding quotient stacks, verifying an instance of a conjecture of Bondal and Orlov.

This https://arxiv.org/abs/2310.01793 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On regular sets in Cayley graphs
Let $\Ga = (V, E)$ be a graph and $a, b$ nonnegative integers. An $(a, b)$-regular set in $\Ga$ is a nonempty proper subset $D$ of $V$ such that every vertex in $D$ has exactly $a$ neighbours in $D$ and every vertex in $V \setminus D$ has exactly $b$ neighbours in $D$. A $(0,1)$-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset $D$ of a group $G$ is called an $(a,b)$-regular set of $G$ if it is an $(a, b)$-regular set in some C…

Bounds and extremal graphs for monitoring edge-geodetic sets in graphs
Florent Foucaud, Pierre-Marie Marcille, Zin Mar Myint, R. B. Sandeep, Sagnik Sen, S. Taruni
https://arxiv.org/abs/2403.09122

Bounds and extremal graphs for monitoring edge-geodetic sets in graphs
A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring edge-geodetic number of $G$, denoted by $meg(G)$, is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem. In this article, we compare $meg(G)$ with some other graph theoretic parameters stemming fr…

I wish we didn’t do this to ourselves.
#yaml

Approximating Small Sparse Cuts
Aditya Anand, Euiwoong Lee, Jason Li, Thatchaphol Saranurak
https://arxiv.org/abs/2403.08983 https://…

Approximating Small Sparse Cuts
We study polynomial-time approximation algorithms for (edge/vertex) Sparsest Cut and Small Set Expansion in terms of $k$, the number of edges or vertices cut in the optimal solution. Our main results are $\mathcal{O}(\text{polylog}\, k)$-approximation algorithms for various versions in this setting. Our techniques involve an extension of the notion of sample sets (Feige and Mahdian STOC'06), originally developed for small balanced cuts, to sparse cuts in general. We then show how to combine t…

This https://arxiv.org/abs/2403.06493 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Lower bound for Secure Domination Number of an Outerplanar Graph
A subset $S$ of vertices in a graph $G$ is a secure dominating set of $G$ if $S$ is a dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup \{u\}$ is also a dominating set of $G$. The secure domination number of $G$, denoted by $γ_{s}(G)$, is the cardinality of a smallest secure dominating sets of $G$. In this paper, we prove that for any outerplanar graph with $n \geq 4$ vertices, $γ_{s}(G) \geq (n+4)/5$ …

Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences
F\'elix Almendra-Hern\'andez, Jes\'us A. De Loera, Sonja Petrovi\'c
https://arxiv.org/abs/2402.09568

Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences
We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-part…