On attraction of slime mould Physarum polycephalum to plants with sedative properties

A plasmodium of acellular slime mould Physarum polycephalum is a large single cell with many nuclei. Presented to a configuration of attracting and repelling stimuli a plasmodium optimizes its growth pattern and spans the attractants, while avoiding repellents, with efficient network of protoplasmic tubes. Such behaviour is interpreted as computation and the plasmodium as an amorphous growing biological computer. Till recently laboratory prototypes of slime mould computing devices (Physarum machines) employed rolled oats and oat powder to represent input data. We explore alternative sources of chemo-attractants, which do not require a sophisticated laboratory synthesis. We show that plasmodium of P. polycephalum prefers sedative herbal tablets and dried plants to oat flakes and honey. In laboratory experiments we develop a hierarchy of slime-mould’s chemo-tactic preferences. We show that Valerian root (Valeriana officinalis) is the strongest chemo-attractant of P. polycephalum outperforming not only most common plants with sedative activities but also some herbal tablets.
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Kernelization and Sparseness: the case of Dominating Set

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be nowhere dense, then we give an almost linear kernel on $\mathcal G$ for the classic Dominating Set problem, i.e., for the case $r=1$. These results generalize a line of previous research on finding linear kernels for Dominating Set and $r$-Dominating Set. However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related Connected Dominating Set problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on $H$-topological-minor-free graphs. Also, we prove that for any somewhere dense class $\mathcal G$, there is some $r$ for which $r$-Dominating Set is W[$2$]-hard on $\mathcal G$. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of $r$-Dominating Set on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded expansion graph

Whole domination in graphs

In this paper, a new parameter of domination number in graphs is defined which is called whole domination number denoted by γwh(G). Some bounds of whole domination number and the number of edges depend on it has been established. Furthermore, the effect of deletion vertex, edge, or add edge have been studied. Also, the effect of the contracting an edge is determined. Finally, some operations between the two graphs have been calculated