Ethanol–Water Separation by Pervaporation through Substituted-Polyacetylene Membranes

Pervaporation; Ethanol–water Mixture; Separation; Membrane; Substituted Polyacetylene; Poly[1-(trimethylsilyl)1-propyne

An NP-Hard Crossing Minimization Problem for Computer Network Layout.

A layout problem of computer communication network is formulated graph-theoretically as follows: "Given a simple graph G = (V,E), find an embedding of G with the minimum number of edge-crossings such that (i) the vertices in V are placed on the circumference of a circle C, and (ii) the edges in E are drawn only inside C." In this paper, this problem will be shown to be NP-hard. Therefore, it is very likely to be intractable

Generation of Maximum Independent Sets of a Biparite Graph and Maximum Cliques of a Circular-Arc Graph.

We present an efflcient algorithm for generating all maximum independent sets of a bipartite graph. Its time complexity is O (n sup 2.5 + (output size )), where n is the number of vertices of a given graph. As its application, we develop an algorithm for generating all maximum cliques of a circular-arc graph. When the graph is given in the form of a family of n arcs on a circle, this algorithm runs in O ( n sup 3.5 + (output size )) time

Efficient Algorithms for Finding Maximum Cliques of an Overlap Graph.

Let F = {I_1, I_2, ..., I_n} be a finite family of closed intervals on the real line. For two distinct intervals I_j and I_k in F, we say that I_j and I_k overlap with each other if they interact with each other but neither one of them contains the other. A graph G = (V, E) is called an overlap graph for F if there is a one-to-one correspondence between V and F such that two vertices in V are adjacent to each other if and only if the corresponding two intervals in F overlap with each other. In this paper, we present two efficient algorithms for finding maximum cliques of an overlap graph when the graph is given in the form of a family of intervals. The first algorithm finds a maximum clique in O time, where n , m are the numbers of vertices , edges, respectively, WEIRD GREEK LETTER is the size of a maximum clique of the graph. The second algorithm generates all maximum cliques of the graph in O time, where WEIRD GREEK LETTER is the total sum of the sizes of the maximum cliques

Efficient Enumeration of Maximal and Maximum Independent Sets of an Interval Graph and a Circular-Arc Graph.

We present efficient algorithms for generating all maximal and all maximum independent sets of an interval graph and a circular- arc graph. When an interval graph is given in the form of a family of n intervals, the first and second algorithms produce all maximal and all maximum independent sets, respectively, in O(n DOT log n+ (the size of an output)) time. When a circular- arc graph is given in the form of a family of n arcs on a circle, the third algorithm generates all maximal independent sets in O(n log n+ (the size of an output)) time. In the same situation, the fourth algorithm enumerates all maximum independent aets in O(n^2+ (the size of an output)) time. The first three algorithms are optimal to within a constant factor

Crossing Minimization in the Straight-Line Embedding of Graphs.

The problem of embedding a graph in the plane with the minimum number of edgecrossings arises in some circuit layout problems. It has been known that this problem is in general NP-hard. An interesting and relevant problem in the area of book embedding was recently shown to be NP-hard. This result implies that the former problem is NP-hard even when the vertices are placed on a straight line l and the edges are drawn completely on either side of l. In this paper, we show that the problem remains NP-hard even if, in addition to these constraints, the positions of the vertices on SCRIPT l are predetermined