A subset
U
of vertices of a graph
G
is called a
determining set
if every automorphism of
G
is uniquely determined by its action on the vertices of
U
. A subset
W
is called a
resolving set
if every ...vertex in
G
is uniquely determined by its distances to the vertices of
W
. Determining (resolving) sets are said to have the
exchange property
in
G
if whenever
S
and
R
are minimal determining (resolving) sets for
G
and
, then there exists
so that
is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchange property. This paper shows that neither determining sets nor resolving sets have the exchange property in all graphs, but that both have the exchange property in trees. It also gives an infinite graph family (
n
-wheels where
n
≥ 8) in which determining sets have the exchange property but resolving sets do not. Further, this paper provides necessary and sufficient conditions for determining sets to have the exchange property in an outerplanar graph.
We define a notion of determining sets for the discrete Laplacian in a domain Ω. A set D is called determining if harmonic functions are uniquely determined by providing their values on D, and if D ...has the same size as the boundary of Ω. It is shown that there exist determining sets that are fairly evenly distributed in Ω. A number of basic properties of determining sets are derived.
We consider a linear space of piecewise polynomials in three variables which are globally smooth, i.e. trivariate
C
1
-splines of arbitrary polynomial degree. The splines are defined on type-6 ...tetrahedral partitions, which are natural generalizations of the four-directional mesh. By using Bernstein–Bézier techniques, we analyze the structure of the spaces and establish formulae for the dimension of the smooth splines on such uniform type partitions.
One of the puzzlingly hard problems in Computer Aided Geometric Design and Approximation Theory is that of finding the dimension of the spline space of
C
r
piecewise degree
n polynomials over a 2D ...triangulation
Ω
. We denote such spaces by
S
n
r
(
Ω
)
. In this note, we restrict
Ω
to have a special structure, namely to be
unconstricted. This will allow for several exact dimension formulas.
Rates of convergence in the central limit theorem are frequently described in terms of the uniform metric. However, statisticians often apply the central limit theorem only at symmetric pairs of ...isolated points, such as the 5% points of the standard normal distribution, ± 1.645. In this paper we study rates of convergence on sets of the form {-θ, θ}, where θ ≥ 0. It is shown that the rate of convergence on the 5% points is the same as the rate uniformly on the whole real line, up to terms of order n-1/2. Curiously, the rate of convergence on the 1% points ± 2.326 can be faster than the rate on the whole real line.
On the determining number of some graphs Afkhami, Mojgan; Amouzegar, Tayyebeh; Khashyarmanesh, Kazem ...
AKCE international journal of graphs and combinatorics,
06/2024
Journal Article
Peer reviewed
Open access
A subset S of vertices of a graph G is a determining set for G if every automorphism of G is uniquely determined by its action on S. The determining number of a graph G is the smallest integer r such ...that G has a determining set of size r. In this paper, we study the determining number of edge-corona product, hierarchical product of graphs and the determining number of blow-up of some graphs. Also, we investigate the determining number of the zero divisor graph of the ring Formula: see text, for some values of n.
A graph
G is said to be
d‐distinguishable if there is a labeling of the vertices with
d labels so that only the trivial automorphism preserves the labels. The smallest such
d is the distinguishing ...number,
Dist
(
G
). A set of vertices
S
⊆
V
(
G
) is a determining set for
G if every automorphism of
G is uniquely determined by its action on
S. The size of a smallest determining set for
G is called the determining number,
Det
(
G
). The orthogonality graph
Ω
2
k has vertices which are bitstrings of length
2
k with an edge between two vertices if they differ in precisely
k bits. This paper shows that
Det
(
Ω
2
k
)
=
2
2
k
−
1 and that, if
m
2
≥
2
k when
k is odd or
m
2
≥
2
k
+
1 when
k is even, then
2
<
Dist
(
Ω
2
k
)
≤
m.
The solution spaces of isogoemetric analysis (IGA) constructed from p degree basis functions allow up to Cp−1 continuity within one patch. However, for a multi-patch domain, the continuity is only C0 ...at the boundaries between the patches. In this study, we present the construction of basis functions of degree p≥2 which are C1 continuous across the common boundaries shared by the patches. The new basis functions are computed as a linear combination of the C0 basis functions on the multi-patch domains. An advantage of the proposed method is that for the new basis functions, the continuity within a patch is preserved, without additional treatment of the functions in the interior of the patch.
We apply continuity constraints to the new basis functions to enforce C1 continuity, where the constraints are developed according to the concept of “matched Gk-constructions always yield Ck-continuous isogeometric elements” discussed in Groisser and Peters, (2015). However, for certain geometries, the over-constrained solution space will lead to C1 locking (Collin and Sangalli, 2016). We discuss and show the usage of partial degree elevation to overcome this problem. We demonstrate the potential of the C1 basis functions for IGA applications through several examples involving biharmonic equations.
•A constructive approach for C1 coupling on 2D and 3D multi-patch domains.•The method is not limited to bilinear parameterizations, but also applicable to general geometries.•The method is compatible with domains containing extraordinary vertices.•Efficiently overcome C1 locking by use of partial degree elevation.
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