scholarly journals Finite skew braces with isomorphic non-abelian characteristically simple additive and circle groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Cindy (Sin Yi) Tsang
Keyword(s):  
Undirected Graph ◽  
Directed Graphs ◽  
Distinguished Vertex

Abstract A skew brace is a triplet ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , where ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are groups such that the brace relation x ∘ ( y ⋅ z ) = ( x ∘ y ) ⋅ x - 1 ⋅ ( x ∘ z ) x\circ(y\cdot z)=(x\circ y)\cdot x^{-1}\cdot(x\circ z) holds for all x , y , z ∈ A x,y,z\in A . In this paper, we study the number of finite skew braces ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , up to isomorphism, such that ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are both isomorphic to T n T^{n} with 𝑇 non-abelian simple and n ∈ N n\in\mathbb{N} . We prove that it is equal to the number of unlabeled directed graphs on n + 1 n+1 vertices, with one distinguished vertex, and whose underlying undirected graph is a tree. In particular, it depends only on 𝑛 and is independent of 𝑇.

scholarly journals Sparse Highly Connected Spanning Subgraphs in Dense Directed Graphs

2018 ◽  
Vol 28 (3) ◽  
pp. 423-464 ◽  
Author(s):  
DONG YEAP KANG
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Analogous Result ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Additional Term ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Polynomial Time Algorithms ◽  
Bipartite Digraph

Mader proved that every strongly k-connected n-vertex digraph contains a strongly k-connected spanning subgraph with at most 2kn - 2k2 edges, where equality holds for the complete bipartite digraph DKk,n-k. For dense strongly k-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly k-connected n-vertex digraph D contains a strongly k-connected spanning subgraph with at most kn + 800k(k + Δ(D)) edges, where Δ(D) denotes the maximum degree of the complement of the underlying undirected graph of a digraph D. Here, the additional term 800k(k + Δ(D)) is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly k-connected n-vertex semicomplete digraph contains a strongly k-connected spanning subgraph with at most kn + 800k2 edges, which is essentially optimal since 800k2 cannot be reduced to the number less than k(k - 1)/2.We also prove an analogous result for strongly k-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.

scholarly journals Intrinsically knotted and 4-linked directed graphs

2018 ◽  
Vol 27 (06) ◽  
pp. 1850037 ◽  
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Edge Contraction ◽  
Special Cases ◽  
Linkless Embedding ◽  
The Relationship ◽  
Knots And Links

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph [Formula: see text] and its corresponding symmetric digraph [Formula: see text]. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is [Formula: see text] in the undirected case, but [Formula: see text] for directed graphs.

scholarly journals Graphs of Morphisms of Graphs

10.37236/919 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
R. Brown ◽  
I. Morris ◽  
J. Shrimpton ◽  
C. D. Wensley
Keyword(s):  
Undirected Graph ◽  
Directed Graphs ◽  
Undirected Graphs ◽  
Single Vertex ◽  
Monoid Structure

This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely 'bands' and 'loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.

Minus Domination Numbers of Directed Graphs

2011 ◽  
Vol 267 ◽  
pp. 334-337
Author(s):  
Wen Sheng Li ◽  
Hua Ming Xing
Keyword(s):  
Lower Bounds ◽  
Directed Graph ◽  
Maximum Degree ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Domination Number ◽  
Directed Cycle ◽  
Domination Numbers

The concept of minus domination number of an undirected graph is transferred to directed graphs. Exact values are found for the directed cycle and particular types of tournaments. Furthermore, we present some lower bounds for minus domination number in terms of the order, the maximum degree, the maximum outdegree and the minimum outdegree of a directed graph.

scholarly journals Symmetrization for Embedding Directed Graphs

2019 ◽  
Vol 33 ◽  
pp. 10043-10044
Author(s):  
Jiankai Sun ◽  
Srinivasan Parthasarathy
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
State Of The Art ◽  
Directed Graphs ◽  
Graph Embedding ◽  
Embedding Problem ◽  
Two Stage ◽  
Second Stage

In this paper, we propose to solve the directed graph embedding problem via a two stage approach: in the first stage, the graph is symmetrized in one of several possible ways, and in the second stage, the so-obtained symmetrized graph is embeded using any state-of-the-art (undirected) graph embedding algorithm. Note that it is not the objective of this paper to propose a new (undirected) graph embedding algorithm or discuss the strengths and weaknesses of existing ones; all we are saying is that whichever be the suitable graph embedding algorithm, it will fit in the above proposed symmetrization framework.

A Browser for Directed Graphs

1984 ◽  
Author(s):  
Lawrence A. Rowe ◽  
Michael Davis ◽  
Eli Messinger ◽  
Carl Meyer ◽  
Charles Spirakis
Keyword(s):  
Directed Graphs

scholarly journals On the star forest polytope for trees and cycles

2019 ◽  
Vol 53 (5) ◽  
pp. 1763-1773
Author(s):  
Meziane Aider ◽  
Lamia Aoudia ◽  
Mourad Baïou ◽  
A. Ridha Mahjoub ◽  
Viet Hung Nguyen
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Discrete Math ◽  
Complete Characterization ◽  
Facial Structure ◽  
Maximum Weight ◽  
Single Node ◽  
End Node ◽  
Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

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