Abstract
We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedlet al. [1]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but \(\mathcal{NP}\)-hard for graphs with maximum degree six. One of our main results is closing the gap in these results, by proving \(\mathcal{NP}\)-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains \(\mathcal{NP}\)-hard for planar graphs with maximum degree six and for 5-regular graphs. On the other hand we present a polynomial time algorithm that determines whether there is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an ‘almost balanced’ ordering.
Supported by grant MEC SB2003-0270. Research completed at the Department of Applied Mathematics and the Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic. Supported by projects LN00A056 and 1M0021620808 of the Ministry of Education of the Czech Republic, and by the European Union Research Training Network COMBSTRU (Combinatorial Structure of Intractable Problems)
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Kára, J., Kratochvíl, J., Wood, D.R. (2005). On the Complexity of the Balanced Vertex Ordering Problem. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_86
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DOI: https://doi.org/10.1007/11533719_86
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28061-3
Online ISBN: 978-3-540-31806-4
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