GD 1999: Graph Drawing pp 225-231

# On 3-Layer Crossings and Pseudo Arrangements

• Imrich Vrt’o
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1731)

## Abstract

Let G = (V 0; V 1; V 2;E) be a 3-layer graph. The 3-layer drawings of G in which V 0, V 1, and V 2 are placed on 3 parallel lines and each edge in E is drawn using one straight line segment, are studied. A generalization of the linear arrangement problem which we call the 3- layer pseudo linear arrangement problem is introduced, and it is shown to be closely related to the 3-layer crossing number. In particular, we show that the 3-layer crossing number of G plus the sum of the square of degrees asymptotically has the same order of magnitude as the optimal solution to the 3-layer linear arrangement problem. Consequently, when G satisfies certain (reasonable) assumptions, we derive the first polynomial time approximation algorithm to compute the 3-layer crossing number within a multiplicative factor of O(log n) from the optimal.

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