Balanced vertex-orderings of graphs
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, a...
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Published in | Discrete Applied Mathematics Vol. 148; no. 1; pp. 27 - 48 |
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Main Authors | , , , , |
Format | Journal Article |
Language | English |
Published |
Lausanne
Elsevier B.V
30.04.2005
Amsterdam Elsevier New York, NY |
Subjects | |
Online Access | Get full text |
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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. |
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ISSN: | 0166-218X 1872-6771 |
DOI: | 10.1016/j.dam.2004.12.001 |