3 Lower Bounds on the Numberings

On Bandwidth, Cutwidth, and Quotient Graphs

The bandwidth and the cutwidth are fundamental parameters which can give indications on the complexity of many problems described in terms of graphs. In this paper, we present a method for nding general upper bounds for the bandwidth and the cutwidth of a given graph from those of any of its quotient graphs. Moreover, general lower bounds are obtained by using vertex-and edge-bisection notions....

Applications of Parameterized st-Orientations in Graph Drawing Algorithms

Many graph drawing algorithms use st-numberings (st-orientations or bipolar orientations) as a first step. An st-numbering of a biconnected undirected graph defines a directed graph with no cycles, one single source s and one single sink t. As there exist exponentially many st-numberings that correspond to a certain undirected graph G, using different st-numberings in various graph drawing algo...

Numberings Optimal for Learning

This paper extends previous studies on learnability in non-acceptable numberings by considering the question: for which criteria which numberings are optimal, that is, for which numberings it holds that one can learn every learnable class using the given numbering as hypothesis space. Furthermore an effective version of optimality is studied as well. It is shown that the effectively optimal num...

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

Lower bounds on the signed (total) $k$-domination number

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...