Abstract
We have introduced the archer's problem and shown that its solution leads to the intersting class of stage graphs which we characterized to be permutation graphs. The characterization which leads to the solution for the archer's problem allowed for the development of improved algorithms for matching in permutation graphs, for a class of two-processor scheduling problems, and for several geometric problems. We answer the natural question of how the archer's problem generalizes to multiple stages and to three-dimensions. In two dimensions we establish upper and lower bounds on the number of stages required to represent graphs. In three dimensions we give characterization results and establish the NP-completeness of the recognition problem already for triangular stages.
There are several interesting open problems suggested by our investigations. The notion of stage number as a graph theoretic parameter seems to be interesting in its own right. This suggests searching for tighter (constructive or not) upper and lower bounds on the stage number of an arbitrary graph, as well as determining the complexity of the recognition problem G ε \(\mathcal{G}\) k , both for fixed as well as variable k.
Research supported in part by NSERC grant.
Research supported in part by ALMERCO Inc.
Work by the author was carried out during a stay at Carleton University.
The full version of this paper is available on the world wide web address http://www/scs.carleton.ca under technical reports.
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Bauernöppel, F. et al. (1995). Optimal shooting: Characterizations and applications. In: Fülöp, Z., Gécseg, F. (eds) Automata, Languages and Programming. ICALP 1995. Lecture Notes in Computer Science, vol 944. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60084-1_76
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DOI: https://doi.org/10.1007/3-540-60084-1_76
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