New models of graph-bin packing.

In Bujtás et al. (2011) 4 the authors introduced a very general problem called Graph-Bin Packing (GBP). It requires a mapping µ : V ( G ) ź V ( H ) from the vertex set of an input graph G into a fixed host graph H, which, among other conditions, satisfies that for each pair u , v of adjacent vertices the distance of µ ( u ) and µ ( v ) in H is between two prescribed bounds. In this paper we propose two online versions of the Graph-Bin Packing problem. In both cases the vertices can arrive in an arbitrary order where each new vertex is adjacent to some of the previous ones. One version is a Maker-Breaker game whose rules are defined by the packing conditions. A subclass of Maker-win input graphs is what we call 'well-packable'; it means that a packing of G is obtained whenever the mapping µ ( u ) is generated by selecting an arbitrary feasible vertex of the host graph for the next vertex of G in each step. The other model is connected-online packing where we are looking for an online algorithm which can always find a feasible packing. In both models we present some sufficient and some necessary conditions for packability. In the connected-online version we also give bounds on the size of used part of the host graph.