scholarly journals New Bounds for Codes Identifying Vertices in Graphs

10.37236/1451 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Gérard Cohen ◽  
Iiro Honkala ◽  
Antoine Lobstein ◽  
Gilles Zémor
Keyword(s):  
Lower Bounds ◽  
Undirected Graph ◽  
Square Lattice ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Two Dimensional ◽  
Identifying Code

Let $G=(V,E)$ be an undirected graph. Let $C$ be a subset of vertices that we shall call a code. For any vertex $v\in V$, the neighbouring set $N(v,C)$ is the set of vertices of $C$ at distance at most one from $v$. We say that the code $C$ identifies the vertices of $G$ if the neighbouring sets $N(v,C), v\in V,$ are all nonempty and different. What is the smallest size of an identifying code $C$ ? We focus on the case when $G$ is the two-dimensional square lattice and improve previous upper and lower bounds on the minimum size of such a code.

SIGNATURE AND RELIABILITY OF CONDITIONAL THREE-DIMENSIONAL CONSECUTIVE-(s, s, s)-OUT-OF-(s, s, m):F SYSTEM

2014 ◽  
Vol 21 (02) ◽  
pp. 1450009 ◽  
Author(s):  
MADHURI G. KULKARNI ◽  
AKANKSHA S. KASHIKAR
Keyword(s):  
Lower Bounds ◽  
Three Dimensional ◽  
Upper And Lower Bounds ◽  
System Failure ◽  
Three Dimension ◽  
Two Dimensional ◽  
Math Model ◽  
The Moment ◽  
Appl Math ◽  
F System

A three-dimensional consecutive (r1, r2, r3)-out-of-(m1, m2, m3):F system was introduced by Akiba et al. [J. Qual. Mainten. Eng.11(3) (2005) 254–266]. They computed upper and lower bounds on the reliability of this system. Habib et al. [Appl. Math. Model.34 (2010) 531–538] introduced a conditional type of two-dimensional consecutive-(r, s)-out-of-(m, n):F system, where the number of failed components in the system at the moment of system failure cannot be more than 2rs. We extend this concept to three dimension and introduce a conditional three-dimensional consecutive (s, s, s)-out-of-(s, s, m):F system. It is an arrangement of ms2 components like a cuboid and it fails if it contains either a cube of failed components of size (s, s, s) or 2s3 failed components. We derive an expression for the signature of this system and also obtain reliability of this system using system signature.

Aprioribounds for a class of nonlinear elliptic equations and applications to physical problems

1981 ◽  
Vol 88 (1-2) ◽  
pp. 75-81 ◽  
Author(s):  
Catherine Bandle
Keyword(s):  
Dirichlet Problem ◽  
Lower Bounds ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Upper And Lower Bounds ◽  
Two Dimensional ◽  
Nonlinear Dirichlet Problem ◽  
Nonlinear Elliptic ◽  
Physical Problems ◽  
Maximal Pressure

SynopsisUpper and lower bounds for the solutions of a nonlinear Dirichlet problem are given and isoperimetric inequalities for the maximal pressure of an ideal charged gas are constructed. The method used here is based on a geometrical result for two-dimensional abstract surfaces.

scholarly journals Minimum survivable graphs with bounded distance increase

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Selma Djelloul ◽  
Mekkia Kouider
Keyword(s):  
Fault Tolerance ◽  
Lower Bounds ◽  
Interconnection Networks ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Np Hard ◽  
Bounded Distance ◽  
Minimum Number ◽  
International Audience

International audience We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.

A remark on positive sojourn times of symmetric processes

2018 ◽  
Vol 55 (1) ◽  
pp. 69-81
Author(s):  
Christophe Profeta
Keyword(s):  
Brownian Motion ◽  
Recurrence Relation ◽  
Lower Bounds ◽  
Sojourn Time ◽  
Occupation Time ◽  
Upper And Lower Bounds ◽  
Stable Processes ◽  
Two Dimensional ◽  
Symmetric Stable Processes ◽  
Sojourn Times

Abstract We show that under some slight assumptions, the positive sojourn time of a product of symmetric processes converges towards ½ as the number of processes increases. Monotony properties are then exhibited in the case of symmetric stable processes, and used, via a recurrence relation, to obtain upper and lower bounds on the moments of the occupation time (in the first and third quadrants) for two-dimensional Brownian motion. Explicit values are also given for the second and third moments in the n-dimensional Brownian case.

scholarly journals Supertrees

10.37236/8971 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Colin Defant ◽  
Noah Kravitz ◽  
Ashwin Sah
Keyword(s):  
Recent Result ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Minimum Length ◽  
Permutation Patterns ◽  
Plane Trees ◽  
Recent Developments ◽  
Pattern Containment ◽  
Significant Attention

A $k$-universal permutation, or $k$-superpermutation, is a permutation that contains all permutations of length $k$ as patterns.  The problem of finding the minimum length of a $k$-superpermutation has recently received significant attention in the field of permutation patterns.  One can ask analogous questions for other classes of objects.  In this paper, we study $k$-supertrees.  For each $d\geq 2$, we focus on two types of rooted plane trees called $d$-ary plane trees and $[d]$-trees.  Motivated by recent developments in the literature, we consider "contiguous" and "noncontiguous" notions of pattern containment for each type of tree.  We obtain both upper and lower bounds on the minimum possible size of a $k$-supertree in three cases; in the fourth, we determine the minimum size exactly.  One of our lower bounds makes use of a recent result of Albert, Engen, Pantone, and Vatter on $k$-universal layered permutations.

scholarly journals Spectral Analysis of Schrödinger Operators with Unusual Semiclassical Behavior

10.14311/1801 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Pavel Exner ◽  
Diana Barseghyan
Keyword(s):  
Spectral Analysis ◽  
Phase Space ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Schrödinger Operators ◽  
Critical Value ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators ◽  
Two Dimensional ◽  
Eigenvalue Bounds

In this paper we discuss several examples of Schrödinger operators describing a particle confined to a region with thin cusp-shaped ‘channels’, given either by a potential or by a Dirichlet boundary; we focus on cases when the allowed phase space is infinite but the operator still has a discrete spectrum. First we analyze two-dimensional operators with the potential |xy|p - ?(x2 + y2)p/(p+2)where p?1 and ??0. We show that there is a critical value of ? such that the spectrum for ??crit it is unbounded from below. In the subcriticalcase we prove upper and lower bounds for the eigenvalue sums. The second part of work is devoted toestimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions havinginfinite cusps which are geometrically nontrivial being either curved or twisted; we are going to showhow these geometric properties enter the eigenvalue bounds.

scholarly journals Maximum difference about the size of optimal identifying codes in graphs differing by one vertex

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mikko Pelto
Keyword(s):  
Upper Bound ◽  
Undirected Graph ◽  
Minimum Size ◽  
Maximum Difference ◽  
Free Graph ◽  
Induced Subgraph ◽  
Identifying Code ◽  
Vertex Set ◽  
International Audience ◽  
Codes In Graphs

Graph Theory International audience Let G=(V,E) be a simple undirected graph. We call any subset C⊆V an identifying code if the sets I(v)={c∈C | {v,c}∈E or v=c } are distinct and non-empty for all vertices v∈V. A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S⊂V is deleted. We provide a tight upper bound for γ(GS)-γ(G) when both graphs are twin-free and |V| is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S|=1.

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