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Balanced vertex-orderings of graphs. (English) Zbl 1060.05088
Summary: In this paper we consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex $$v$$ are as evenly distributed to the left and right of $$v$$ as possible. This problem, which has applications in graph drawing for example, is shown to be NP-hard, and remains NP-hard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertex-ordering, obtaining optimal orderings for directed acyclic graphs, trees, and graphs with maximum degree three. For undirected graphs, we obtain a 13/8-approximation algorithm. Finally we consider the problem of determining a balanced vertex-ordering of a bipartite graph with a fixed ordering of one bipartition. When only the imbalances of the fixed vertices count, this problem is shown to be NP-hard. On the other hand, we describe an optimal linear time algorithm when the final imbalances of all vertices count. We obtain a linear time algorithm to compute an optimal vertex-ordering of a bipartite graph with one bipartition of constant size.

##### MSC:
 05C85 Graph algorithms (graph-theoretic aspects) 05C10 Planar graphs; geometric and topological aspects of graph theory 68R10 Graph theory (including graph drawing) in computer science
##### Keywords:
Graph algorithm; Graph drawing; Vertex-ordering; Balanced
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