On the Convexity of Independent Set Games
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This paper investigates independent set games (introduced by Deng et al., Math. Oper. Res., 24:751-766, 1999), which belong to cooperative profit games. Let G = (V,E) be an undirected graph and α(G) be the size of maximum independent sets in G. For any F ⊆ E, V 〈 F 〉 denotes the set of vertices incident only to edges in F, and G[V 〈 F 〉] denotes the induced subgraph on V 〈 F〉. An independent set game on G is a cooperative game Γ_G = (E, γ), where E is the set of players and γ : 2^E →R is the characteristic function such that γ(F) = α(G[V 〈 F 〉 ]) for any F ⊆ E. Independent set games were first studied by Deng et al., where the algorithmic aspect of the core was investigated and a complete characterization for the core non-emptiness was presented. In this paper, we focus on the convexity of independent set games, since convex games possess many nice properties both economically and computationally. We propose the first complete characterization for the convexity of independent set games, i.e., every non-pendant edge is incident to a pendant edge in the underlying graph. Our characterization immediately yields a polynomial time algorithm for recognizing convex instances of independent set games. We also introduce two relaxations of independent set games and characterize their convexity respectively.
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