Separator-based graph embedding into multidimensional grids with small edge-congestion

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a well-known notion of graph separators, we show that an embedding with a smaller edge-congestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prove that any graph with NN nodes, maximum node degree ΔΔ, and with a node-separator of size ss, where ss is a function such that s(n)=O(nα)s(n)=O(nα) with 0≤α<10≤α<1, can be embedded into a grid of a fixed dimension d≥2d≥2 with at least NN nodes, with an edge-congestion of O(Δ)O(Δ) if d>1/(1−α)d>1/(1−α), O(ΔlogN)O(ΔlogN) if d=1/(1−α)d=1/(1−α), and O(ΔNα−1+1d) if d<1/(1−α)d<1/(1−α). This edge-congestion achieves constant ratio approximation if d>1/(1−α)d>1/(1−α), and matches an existential lower bound within a constant factor if d≤1/(1−α)d≤1/(1−α). Our result implies that if the guest graph has an excluded minor of a fixed size, such as a planar graph, then we can obtain an edge-congestion of O(ΔlogN)O(ΔlogN) for d=2d=2 and O(Δ)O(Δ) for any fixed d≥3d≥3. Moreover, if the guest graph has a fixed treewidth, such as a tree, an outerplanar graph, and a series–parallel graph, then we can obtain an edge-congestion of O(Δ)O(Δ) for any fixed d≥2d≥2. To design our embedding algorithm, we introduce edge-separators bounding extension  , such that in partitioning a graph into isolated nodes using edge-separators recursively, the number of outgoing edges from a subgraph to be partitioned in a recursive step is bounded. We present an algorithm to construct an edge-separator with extension of O(Δnα)O(Δnα) from a node-separator of size O(nα)O(nα).