scholarly journals Isotropic Matroids III: Connectivity

10.37236/5937 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Pendant Vertex ◽  
Split Decomposition ◽  
Binary Matroid ◽  
Isotropic System ◽  
Interesting Theorem

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically $n$-connected. This abstract-seeming result is equivalent to the more concrete assertion that $G$ is locally equivalent to a graph with a vertex of degree $<\frac{n-1}{2}$.

scholarly journals A Characterization of Circle Graphs in Terms of Multimatroid Representations

10.37236/6992 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Simple Graph ◽  
Rank Function ◽  
Binary Matroid ◽  
Circle Graph ◽  
Circle Graphs ◽  
Isotropic System

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

scholarly journals On Binary Matroid Minors and Applications to Data Storage over Small Fields

2017 ◽  
pp. 139-153
Author(s):  
Matthias Grezet ◽  
Ragnar Freij-Hollanti ◽  
Thomas Westerbäck ◽  
Camilla Hollanti
Keyword(s):  
Data Storage ◽  
Binary Matroid ◽  
Small Fields

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

scholarly journals Molecular analysis of the dengue virus type 1 and 2 in Brazil based on sequences of the genomic envelope-nonstructural protein 1 junction region

2004 ◽  
Vol 46 (3) ◽  
pp. 145-152 ◽  
Author(s):  
Cecília Luiza S. Santos ◽  
Maria Anice M. Sallum ◽  
Peter G. Foster ◽  
Iray Maria Rocco
Keyword(s):  
Sequence Data ◽  
Nonstructural Protein ◽  
Phylogenetic Analyses ◽  
Decomposition Analysis ◽  
Structural Protein ◽  
Network Graph ◽  
Junction Region ◽  
Split Decomposition ◽  
Genotype Iii ◽  
Nonstructural Protein 1

The genomic sequences of the Envelope-Non-Structural protein 1 junction region (E/NS1) of 84 DEN-1 and 22 DEN-2 isolates from Brazil were determined. Most of these strains were isolated in the period from 1995 to 2001 in endemic and regions of recent dengue transmission in São Paulo State. Sequence data for DEN-1 and DEN-2 utilized in phylogenetic and split decomposition analyses also include sequences deposited in GenBank from different regions of Brazil and of the world. Phylogenetic analyses were done using both maximum likelihood and Bayesian approaches. Results for both DEN-1 and DEN-2 data are ambiguous, and support for most tree bipartitions are generally poor, suggesting that E/NS1 region does not contain enough information for recovering phylogenetic relationships among DEN-1 and DEN-2 sequences used in this study. The network graph generated in the split decomposition analysis of DEN-1 does not show evidence of grouping sequences according to country, region and clades. While the network for DEN-2 also shows ambiguities among DEN-2 sequences, it suggests that Brazilian sequences may belong to distinct subtypes of genotype III.

An algebraic characterization of indistinguishable cardinals

1970 ◽  
Vol 35 (1) ◽  
pp. 97-104
Author(s):  
A. B. Slomson
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Algebraic Characterization ◽  
Algebraic Criterion ◽  
Characterizable Cardinals ◽  
Second Order Logic ◽  
Interesting Theorem ◽  
Straightforward Proof

Two cardinals are said to beindistinguishableif there is no sentence of second order logic which discriminates between them. This notion, which is defined precisely below, is closely related to that ofcharacterizablecardinals, introduced and studied by Garland in [3]. In this paper we give an algebraic criterion for two cardinals to be indistinguishable. As a consequence we obtain a straightforward proof of an interesting theorem about characterizable cardinals due to Zykov [6].

scholarly journals On the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertex

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
I. H. Agustin ◽  
M. I. Utoyo ◽  
Dafik ◽  
M. Venkatachalam ◽  
Surahmat
Keyword(s):  
Natural Number ◽  
Finite Graph ◽  
Star Graph ◽  
Pendant Vertex ◽  
Vertex Set

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

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