Convexity of graph-restricted games induced by minimum partitions
RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 3, pp. 841-866.

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components corresponding to a minimum partition. This minimum partition 𝒫min is i nduced by the deletion of the minimum weight edges. We provide five necessary conditions on the graph edge-weights to have inheritance of convexity from the underlying game to the restricted game associated with 𝒫min. Then, we establish that these conditions are also sufficient for a weaker condition, called ℱ-convexity, obtained by restriction of convexity to connected subsets. Moreover, we prove that inheritance of convexity for Myerson restricted game associated with a given graph G is equivalent to inheritance of ℱ-convexity for the 𝒫min-restricted game associated with a particular weighted graph G′ built from G by adding a dominating vertex, and with only two different edge-weights. Then, we prove that G is cycle-complete if and only if a specific condition on adjacent cycles is satisfied on G′.

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Accepté le :
DOI : 10.1051/ro/2017069
Classification : 91A12, 91A43, 90C27, 05C75
Mots-clés : Communication networks, cooperative game, convex game, restricted game, partitions
Skoda, Alexandre 1

1 
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     title = {Convexity of graph-restricted games induced by minimum partitions},
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Skoda, Alexandre. Convexity of graph-restricted games induced by minimum partitions. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 3, pp. 841-866. doi : 10.1051/ro/2017069. http://www.numdam.org/articles/10.1051/ro/2017069/

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