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The Computer Journal Advance Access originally published online on January 9, 2007
The Computer Journal 2007 50(3):357-368; doi:10.1093/comjnl/bxl068
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© The Author 2007. Published by Oxford University Press on behalf of The British Computer Society. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Edge-Orienting on Split, Planar and Treelike Graphs

William Chung-Kung Yen*

Department of Information Management, Shih Hsin University, Taipei, 116 Taiwan

* Corresponding author: ckyen001{at}ms7.hinet.net

Received 3 January 2006; revised 2 November 2006

Let G(V, E) be an undirected connected graph, where each vertex v is associated with a positive cost C(v) and each edge e = (u, v) is associated with two positive weights, W(u -> v) and W(v -> u). We consider a new graph problem, called the edge-orientation problem (the EOP). The major issue is to assign each edge e = (u, v) an orientation, either from u to v, denoted as u -> v, or from v to u, denoted as v -> u, such that maxxisinV{C(x) + {Sigma}x->z W(x -> z)} is minimized. This paper first shows that the EOP is NP-hard on split graphs and planar graphs. Then, a linear-time algorithm on star graphs is proposed by the prune-and-search strategy. Finally, the algorithmic result on star graphs is extended to trees and simple cactus graphs using the dynamic programming strategy.

Key Words: The edge-orientation problem • split graphs • planar graphs • treelike graphs • prune-and-search strategy • dynamic programming strategy


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