Yannakakis’ Clique versus Independent Set problem (CL− IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size O(n), and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a p...

A superfine Li0.33La0.557TiO₃ (LLTO, 69.4 nm) was successfully synthesized by a facile solvent-thermal method to enhance the electrochemical properties of the lithium-ion battery separator. Co-axial nanofiber of cellulose and Poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF-HFP) was prepared by a co-axial electrospinning technique, in which the shell material was PVDF-HFP and the core was...

In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K ∩S = ∅, there exists a cut (W,W ′) in C such that K ⊆W and S ⊆W ′. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [20] asked in 1...

We show how to construct an O p n separator decomposition of a planar graph G in O n time Such a decomposition de nes a binary tree where each node corresponds to a subgraph of G and stores an O p n separator of that subgraph We also show how to construct an O n way decomposition tree in parallel in O log n time so that each node corresponds to a subgraph of G and stores an O n separator of tha...

The combined use of brush anodes and glass fiber (GF1) separators, and plastic mesh supporters were used here for the first time to create a scalable microbial fuel cell architecture. Separators prevented short circuiting of closely-spaced electrodes, and cathode supporters were used to avoid water gaps between the separator and cathode that can reduce power production. The maximum power densit...

Acoustic cell retention devices have provided a practical alternative for up to 50 L/day perfusion cultures but further scale-up has been limited. A novel temperature-controlled and larger-scale acoustic separator was evaluated at up to 400 L/day for a 10(7) CHO cell/mL perfusion culture using a 100-L bioreactor that produced up to 34 g/day recombinant protein. The increased active volume of th...

Plotkin, Rao, and Smith (SODA’97) showed that any graph with m edges and n vertices that excludes Kh as a depth O(l logn)-minor has a separator of size O(n/l + lh logn) and that such a separator can be found in O(mn/l) time. A time bound of O(m + n/l) for any constant ǫ > 0 was later given (W., FOCS’11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has t...

A set of vertices S ⊆ V is called a safe separator for treewidth, if S is a separator of G, and the treewidth of G equals the maximum of the treewidth over all connected components W of G S of the graph, obtained by making S a clique in the subgraph of G, induced by W ∪ S. We show tha t such safe separators are a very powerful tool for preprocessing graphs when we want to compute their treewidt...