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We examine cooperative games where the viability of a coalition is determinedby whether or not its members have the ability to communicate amongstthemselves independently of non-members. This necessary condition for viabilitywas proposed by Myerson (1977) and is modeled via an interaction graph$G=(V,E)$; a coalition $S\subseteq V$ is then viable if and only if the inducedgraph $G[S]$ is connected. The non-emptiness of the core of a coalition gamecan be tested by a well-known covering LP. Moreover, the integrality gap of itsdual packing LP defines exactly the multiplicative least-core and the relativecost of stability of the coalition game. This gap is upper bounded by thepacking-covering ratio which, for graphical coalition games, is known to be atmost the treewidth of the interaction graph plus one (Meir et al. 2013).  We examine the packing-covering ratio and integrality gaps of graphicalcoalition games in more detail. We introduce the thicket parameter of a graph,and prove it precisely measures the packing-covering ratio. It alsoapproximately measures the primal and dual integrality gaps. The thicket numberprovides an upper bound of both integrality gaps. Moreover we show that for anyinteraction graph, the primal integrality gap is, in the worst case, linear interms of the thicket number while the dual integrality gap is polynomial interms of it. At the heart of our results, is a graph theoretic minmax theoremshowing the thicket number is equal to the minimum width of a vinedecomposition of the coalition graph (a vine decomposition is a generalizationof a tree decomposition). We also explain how the thicket number relates to theVC-dimension of the set system produced by the game.

Coalition Games on Interaction Graphs: A Horticultural Perspective