Balanced vertex-orderings of graphs

• Published in: Discrete Applied Mathematics Vol. 148; no. 1; pp. 27 - 48
• Main Authors: Biedl, Therese, Chan, Timothy, Ganjali, Yashar, Hajiaghayi, Mohammad Taghi, Wood, David R.
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 30.04.2005; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Balanced; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graph algorithm; Graph drawing; Graph theory; Information retrieval. Graph; Mathematics; Sciences and techniques of general use; Theoretical computing; Vertex-ordering; Graph drawing; Vertex-ordering; Graph algorithm; Balanced; Tree(graph); Computer theory; Optimal algorithm; Vertex ordering; Approximation algorithm; Graph degree; Bipartite graph; Combinatorics; Acyclic graph; Directed graph
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2004.12.001

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.