Arbitrary Overlap Constraints in Graph Packing Problems

In earlier versions of the community discovering problem, the overlap between communities was restricted by a simple count upper-bound. In this paper, we introduce the [Formula: see text]-Packing with [Formula: see text]-Overlap problem to allow for more complex constraints in the overlap region than those previously studied. Let [Formula: see text] be all possible subsets of vertices of [Formula: see text] each of size at most [Formula: see text], and [Formula: see text] be a function. The [Formula: see text]-Packing with [Formula: see text]-Overlap problem seeks at least [Formula: see text] induced subgraphs in a graph [Formula: see text] subject to: (i) each subgraph has at most [Formula: see text] vertices and obeys a property [Formula: see text], and (ii) for any pair [Formula: see text], with [Formula: see text], [Formula: see text] (i.e., the pair [Formula: see text] does not conflict). We also consider a variant that arises in clustering applications: each subgraph of a solution must contain a set of vertices from a given collection of sets [Formula: see text], and no pair of subgraphs may share vertices from the sets of [Formula: see text]. In addition, we propose similar formulations for packing hypergraphs. We give an [Formula: see text] algorithm for our problems where [Formula: see text] is the parameter and [Formula: see text] and [Formula: see text] are constants, provided that: (i) [Formula: see text] is computable in polynomial time in [Formula: see text] and (ii) the function [Formula: see text] satisfies specific conditions. Specifically, [Formula: see text] is hereditary, applicable only to overlapping subgraphs, and computable in polynomial time in [Formula: see text] and [Formula: see text]. Motivated by practical applications we give several examples of [Formula: see text] functions which meet those conditions.

Efficient Graph Packing via Game Colouring

The game colouring number gcol(G) of a graphGis the leastksuch that, if two players take turns choosing the vertices of a graph, then either of them can ensure that every vertex has fewer thankneighbours chosen before it, regardless of what choices the other player makes. Clearly gcol(G) ≤ Δ(G)+1. Sauer and Spencer [20] proved that if two graphsG1andG2onnvertices satisfy 2Δ(G1)Δ(G2) <nthen they pack,i.e., there is an embedding ofG1into the complement ofG2. We improve this by showing that if (gcol(G1)−1)Δ(G2)+(gcol(G2)−1)Δ(G1) <nthenG1andG2pack. To our knowledge this is the first application of colouring games to a non-game problem.