scholarly journals Blockers for Triangulations of a Convex Polygon and a Geometric Maker-Breaker Game

10.37236/7205 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Chaya Keller ◽  
Yael Stein
Keyword(s):  
Convex Polygon ◽  
Spanning Trees ◽  
Fibonacci Sequence ◽  
Geometric Graph ◽  
Complete Characterization ◽  
Common Edge ◽  
Hamiltonian Paths ◽  
The Family ◽  
Vertex Set

 Let $G$ be a complete convex geometric graph whose vertex set $P$ forms a convex polygon $C$, and let $\mathcal{F}$ be a family of subgraphs of $G$. A blocker for $\mathcal{F}$ is a set of diagonals of $C$, of smallest possible size, that contains a common edge with every element of $\mathcal{F}$. Previous works determined the blockers for various families $\mathcal{F}$ of non-crossing subgraphs, including the families of all perfect matchings, all spanning trees, all Hamiltonian paths, etc. In this paper we present a complete characterization of the family $\mathcal{B}$ of blockers for the family $\mathcal{T}$ of triangulations of $C$. In particular, we show that $|\mathcal{B}|=F_{2n-8}$, where $F_k$ is the $k$'th element in the Fibonacci sequence and $n=|P|$. We use our characterization to obtain a tight result on a geometric Maker-Breaker game in which the board is the set of diagonals  of a convex $n$-gon $C$ and Maker seeks to occupy a triangulation of $C$. We show that in the $(1:1)$ triangulation game, Maker can ensure  a win within $n-3$ moves, and that in the $(1:2)$ triangulation game, Breaker can ensure a win within $n-3$ moves. In particular, the threshold bias for the game is $2$.

Transformations preserving the Lyapunov exponents

2018 ◽  
Vol 20 (04) ◽  
pp. 1750027 ◽  
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Lyapunov Exponents ◽  
Asymptotic Properties ◽  
Regular Sequence ◽  
Complete Characterization ◽  
Block Form ◽  
The Family ◽  
Coordinate Changes ◽  
Invertible Matrices ◽  
Finite Exponent

We give a complete characterization of the existence of Lyapunov coordinate changes bringing an invertible sequence of matrices to one in block form. In other words, we give a criterion for the block-trivialization of a nonautonomous dynamics with discrete time while preserving the asymptotic properties of the dynamics. We provide two nontrivial applications of this criterion: we show that any Lyapunov regular sequence of invertible matrices can be transformed by a Lyapunov coordinate change into a constant diagonal sequence; and we show that the family of all coordinate changes preserving simultaneously the Lyapunov exponents of all sequences of invertible matrices (with finite exponent) coincides with the family of Lyapunov coordinate changes.

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals Graph Reconstruction from Unlabeled Edge Lengths

2021 ◽  
Author(s):  
Dániel Garamvölgyi ◽  
Tibor Jordán
Keyword(s):  
Complete Characterization ◽  
Partial Characterization ◽  
Reconstruction Problem ◽  
Global Rigidity ◽  
Generic Framework ◽  
Graph Reconstruction ◽  
The Family ◽  
Rigidity Property ◽  
Low Dimensional

AbstractA d-dimensional framework is a pair (G, p), where $$G=(V,E)$$ G = ( V , E ) is a graph and p is a map from V to $$\mathbb {R}^d$$ R d . The length of an edge $$uv\in E$$ u v ∈ E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in $$\mathbb {R}^d$$ R d if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent paper Gortler, Theran, and Thurston proved that if every generic framework (G, p) in $$\mathbb {R}^d$$ R d is globally rigid for some graph G on $$n\ge d+2$$ n ≥ d + 2 vertices (where $$d\ge 2$$ d ≥ 2 ), then already the set of (unlabeled) edge lengths of a generic framework (G, p), together with n, determine the framework up to congruence. In this paper we investigate the corresponding unlabeled reconstruction problem in the case when the above generic global rigidity property does not hold for the graph. We provide families of graphs G for which the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of vertices, uniquely determine the graph, up to isomorphism. We call these graphs weakly reconstructible. We also introduce the concept of strong reconstructibility; in this case the labeling of the edges is also determined by the set of edge lengths of any generic framework. For $$d=1,2$$ d = 1 , 2 we give a partial characterization of weak reconstructibility as well as a complete characterization of strong reconstructibility of graphs. In particular, in the low-dimensional cases we describe the family of weakly reconstructible graphs that are rigid but not redundantly rigid.

Structure and properties of strong prefix codes of pictures

2015 ◽  
Vol 27 (2) ◽  
pp. 123-142 ◽  
Author(s):  
MARCELLA ANSELMO ◽  
DORA GIAMMARRESI ◽  
MARIA MADONIA
Keyword(s):  
Polynomial Time ◽  
Complete Characterization ◽  
Structure And Properties ◽  
Minimal Size ◽  
Prefix Code ◽  
Decoding Algorithm ◽  
Prefix Codes ◽  
The Family ◽  
Definition Of

A setX⊆ Σ** of pictures is a code if every picture over Σ is tilable in at most one way with pictures inX. The definition ofstrong prefix codeis introduced. The family of finite strong prefix codes is decidable and it has a polynomial time decoding algorithm. Maximality for finite strong prefix codes is also studied and related to the notion of completeness. We prove that any finite strong prefix code can be embedded in a unique maximal strong prefix code that has minimal size and cardinality. A complete characterization of the structure of maximal finite strong prefix codes completes the paper.

scholarly journals A Classification of Ramanujan Unitary Cayley Graphs

10.37236/478 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Andrew Droll
Keyword(s):  
Cayley Graph ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Complete Characterization ◽  
Absolute Value ◽  
Ramanujan Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

The unitary Cayley graph on $n$ vertices, $X_n$, has vertex set ${\Bbb Z}/{n\Bbb Z}$, and two vertices $a$ and $b$ are connected by an edge if and only if they differ by a multiplicative unit modulo $n$, i.e. ${\rm gcd}(a-b,n) = 1$. A $k$-regular graph $X$ is Ramanujan if and only if $\lambda(X) \leq 2\sqrt{k-1}$ where $\lambda(X)$ is the second largest absolute value of the eigenvalues of the adjacency matrix of $X$. We obtain a complete characterization of the cases in which the unitary Cayley graph $X_n$ is a Ramanujan graph.

Bisphosphine ligands containing two o-N,N-dimethylanilinyl substituents at each phosphorus atom

2002 ◽  
Vol 80 (11) ◽  
pp. 1600-1606 ◽  
Author(s):  
Nathan D Jones ◽  
Patric Meessen ◽  
Martin B Smith ◽  
Udo Losehand ◽  
Steven J Rettig ◽  
...  
Keyword(s):  
Crystal Structure ◽  
Hydrogen Bonding ◽  
Hydrogen Bonds ◽  
Structure Data ◽  
Phosphorus Atom ◽  
Crystal Structure Data ◽  
Complete Characterization ◽  
The Family ◽  
Weak Hydrogen Bonds

The synthesis and complete characterization of the family of tetra(amine)bisphosphine ligands (o-NMe2C6H4)2P-(X)-P(o-NMe2C6H4)2, where X = CH2 (dmapm), (CH2)2 (dmape), and [Formula: see text] (dmapcp), are described. Crystal structure data are compared with known, analogous bisphosphines containing o-pyridyl or phenyl substituents in place of the o-dimethylanilinyl groups. Several short, intramolecular C-H···N distances in the anilinyl derivatives may represent the presence of weak hydrogen bonds. Key words: phosphine, amine, polydentate, hydrogen-bonding to N atoms.

scholarly journals Nowhere-Zero 3-Flows in Squares of Graphs

10.37236/1698 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Rui Xu ◽  
Cun-Quan Zhang
Keyword(s):  
Connected Graph ◽  
Complete Characterization ◽  
The Family

It was conjectured by Tutte that every 4-edge-connected graph admits a nowhere-zero $3$-flow. In this paper, we give a complete characterization of graphs whose squares admit nowhere-zero $3$-flows and thus confirm Tutte's $3$-flow conjecture for the family of squares of graphs.

Prothrombin Metz : Purification and Characterization of a Variant of Human Prothrombin

1979 ◽  
Author(s):  
M Ribieto ◽  
J Elion ◽  
D Labie ◽  
F Josso
Keyword(s):  
Molecular Weight ◽  
Gel Electrophoresis ◽  
Polyacrylamide Gel Electrophoresis ◽  
Factor Xa ◽  
Polyacrylamide Gel ◽  
Cleavage Pattern ◽  
Purification And Characterization ◽  
Double Immunodiffusion ◽  
The Family

For the purification of the abnormal prothrombin (Pt Metz), advantage has been taken of the existence in the family of three siblings who, being double heterozygotes for Pt Metz and a hypoprothrombinemia, have no normal Pt. Purification procedures included barium citrate adsorption and chromatography on DEAE Sephadex as for normal Pt. As opposed to some other variants (Pt Barcelona and Madrid), Pt Metz elutes as a single symetrical peak. By SDS polyacrylamide gel electrophoresis, this material is homogeneous and appears to have the same molecular weight as normal Pt. Comigration of normal and abnormal Pt in the absence of SDS, shows a double band suggesting an abnormal charge for the variant. Pt Metz exhibits an identity reaction with the control by double immunodiffusion. Upon activation by factor Xa, Pt Metz can generate amydolytic activity on Bz-Phe-Val-Arg-pNa (S2160), but only a very low clotting activity. Clear abnormalities are observed in the cleavage pattern of Pt Metz when monitored by SDS gel electrophoresis. The main feature are the accumulation of prethrombin l (Pl) and the appearance of abnormal intermediates migrating faster than Pl.

Experimental Characterization of Mechanical Behavior of Cord-Rubber Composites

1982 ◽  
Vol 10 (1) ◽  
pp. 37-54 ◽  
Author(s):  
M. Kumar ◽  
C. W. Bert
Keyword(s):  
Response Characteristic ◽  
Cord Compression ◽  
Complete Characterization ◽  
Strain Response ◽  
Strain Gages ◽  
Experimental Characterization ◽  
Rubber Composites ◽  
Stress Strain ◽  
Strain Data

Abstract Unidirectional cord-rubber specimens in the form of tensile coupons and sandwich beams were used. Using specimens with the cords oriented at 0°, 45°, and 90° to the loading direction and appropriate data reduction, we were able to obtain complete characterization for the in-plane stress-strain response of single-ply, unidirectional cord-rubber composites. All strains were measured by means of liquid mercury strain gages, for which the nonlinear strain response characteristic was obtained by calibration. Stress-strain data were obtained for the cases of both cord tension and cord compression. Materials investigated were aramid-rubber, polyester-rubber, and steel-rubber.

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