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A variation on the min cut linear arrangement problem. (English) Zbl 0643.68094
Summary: The min cut linear arrangement problem asks, for a given graph G and a positive integer k, if there exists a linear arrangement of G’s vertices so that any line separating consecutive vertices in the layout cuts at most k of the edges. A variation of this problem insists that the arrangement be made on a (fixed-degree) tree instead of a line. We show that (1) this problem is NP-complete even when G is planar; (2) it is easily solved when G is a tree; and (3) there is a simple characterization for all graphs with cost 2 or less. Our main result is a linear-time algorithm to embed an outerplanar graph G into a spanning tree with cost at most max degree(G)$$+1$$. This result is important because it extends to an approximation algorithm for the standard min cut linear arrangement problem on outerplanar graphs.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 68Q25 Analysis of algorithms and problem complexity 05C10 Planar graphs; geometric and topological aspects of graph theory
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