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A Graph-Grabbing Game

Published online by Cambridge University Press:  09 March 2011

PIOTR MICEK
Affiliation:
Department of Theoretical Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland (e-mail: Piotr.Micek@tcs.uj.edu.pl, Bartosz.Walczak@tcs.uj.edu.pl, URL: http://tcs.uj.edu.pl/Micek, http://tcs.uj.edu.pl/Walczak)
BARTOSZ WALCZAK
Affiliation:
Department of Theoretical Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland (e-mail: Piotr.Micek@tcs.uj.edu.pl, Bartosz.Walczak@tcs.uj.edu.pl, URL: http://tcs.uj.edu.pl/Micek, http://tcs.uj.edu.pl/Walczak)

Abstract

Two players share a connected graph with non-negative weights on the vertices. They alternately take the vertices (one in each turn) and collect their weights. The rule they have to obey is that the remaining part of the graph must be connected after each move. We conjecture that the first player can get at least half of the weight of any tree with an even number of vertices. We provide a strategy for the first player to get at least 1/4 of an even tree. Moreover, we confirm the conjecture for subdivided stars. The parity condition is necessary: Alice gets nothing on a three-vertex path with all the weight at the middle. We suspect a kind of general parity phenomenon, namely, that the first player can gather a substantial portion of the weight of any ‘simple enough’ graph with an even number of vertices.

Type
Paper
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Cibulka, J., Kynčl, J., Mészáros, V., Stolař, R. and Valtr, P. (2009) Solution to Peter Winkler's pizza problem. In Combinatorial Algorithms (Fiala, J., Kratochvíl, J. and Miller, M., eds), Vol. 5874 of Lecture Notes in Computer Science, Springer, pp. 356367.CrossRefGoogle Scholar
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