We develop graph theoretic methods for analyzing maximally entangled pure states distributed between a number of different parties. We introduce a technique called bicolored merging, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication. We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among n agents. Further, we discuss a new graph theoretic metric on a class of multipartite states, and its implications.
We extend the concept of the polygon visible from a source point S in a simple polygon by considering visibility with two types of reflection, specular and diffuse. In specular reflection a light ray reflects from an edge of the polygon according to the rule: the angle of incidence equals the angle of reflection. In diffuse reflection a light ray reflects from an edge of the polygon in all inward directions. Several geometric and combinatorial properties of visibility polygons under these two types of reflection are described, when at most one reflection is permitted. We show that the visibility polygon Vs(S) under specular reflection may be nonsimple, while the visibility polygon Vd(S) under diffuse reflection is always simple. We present a (n 2 ) worst-case bound on the combinatorial complexity of both Vs(S) and Vd(S) and describe simple O(n 2 log 2 n) time algorithms for constructing the sets.
In this paper we propose a general framework for viewing a class of heuristics for track assignment in channel routing from a purely graph theoretic angle. Within this framework we propose algorithms for computing roufing solutions using optimal or near optimal number of tracks for several well-known benchmark channels in the two-layer VH, three-layer HVH, and multi-layer Vi Hi and Vi Hi+l routing models. Within the same framework we also design an algorithm for minimizing the total wire length in the two-layer VH and three-layer H V H routing models.
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