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Domination Subdivision Numbers

100%

Haynes T., Hedetniemi S., Hedetniemi S., Jacobs D., Knisely J., van der Merwe L.

Discussiones Mathematicae Graph Theory

2001

tom 21

nr 2

239-253

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $sd_γ(G)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that $1 ≤ sd_γ(G) ≤ 3$ for any graph G. We give a counterexample to this conjecture. On the other hand, we show that $sd_γ(G) ≤ γ(G)+1$ for any graph G without isolated vertices, and give constant upper bounds on $sd_γ(G)$ for several families of graphs.

Ternary wavelets and their applications to signal compression

80%

Mustafa G., Chen F., Huang Z.

International Journal of Applied Mathematics and Computer Science

2004

tom 14

nr 2

233-240

We introduce ternary wavelets, based on an interpolating 4-point C^2 ternary stationary subdivision scheme, for compressing fractal-like signals. These wavelets are tightly squeezed and therefore they are more suitable for compressing fractal-like signals. The error in compressing fractal-like signals by ternary wavelets is at most half of that given by four-point wavelets (Wei and Chen, 2002). However, for compressing regular signals we further classify ternary wavelets into 'odd ternary' and 'even ternary' wavelets. Our odd ternary wavelets are better in part for compressing both regular and fractal-like signals than four-point wavelets. These ternary wavelets are locally supported, symmetric and stable. The analysis and synthesis algorithms have linear time complexity.

Fundamental pro-groupoids and covering projections

41%

Hernández-Paricio L. J.

Fundamenta Mathematicae

1998

tom 156

nr 1

1-31

We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX, Sets), where π CX is the Čech fundamental pro-groupoid of X. If X is locally path-connected and semilocally 1-connected, we show that π crs (X) is weakly equivalent to π X, the standard fundamental groupoid of X, and in this case pro (π crs (X), Sets) is equivalent to the functor category $Sets^{π X}$. If (X,*) is a pointed connected compact metrisable space and if (X,*) is 1-movable, then the category of covering projections of X is equivalent to the category of continuous $\check π_1 (X,*)$-sets, where $\check π_1 (X,*)$ is the Čech fundamental group provided with the inverse limit topology.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Fundamenta Mathematicae

1 International Journal of Applied Mathematics and Computer Science

1 Chen F.

1 Haynes T.

1 Hedetniemi S.

1 Hernández-Paricio L. J.

1 Huang Z.

1 Jacobs D.

1 Knisely J.

1 Mustafa G.

1 van der Merwe L.

1 2004

1 2001

1 1998

Wyniki wyszukiwania

Domination Subdivision Numbers

Ternary wavelets and their applications to signal compression

Fundamental pro-groupoids and covering projections