Computing the Stretch of an Embedded Graph

Abstract

Let G be a graph embedded in an orientable surface Σ, possibly with edge weights, and denote by len(γ) the length (the number of edges or the sum of the edge weights) of a cycle γ in G. The stretch of a graph embedded on a surface is the minimum of len(α) • len(β) over all pairs of cycles α and β that cross exactly once. We provide an algorithm to compute the stretch of an embedded graph in time O(g 4 n log n) with high probability, or in time O(g 4 n log 2 n) in the worst case, where g is the genus of the surface Σ and n is the number of vertices in G.

Toroidal grid minors and stretch in embedded graphs

“…In the quest for another embedding density parameter suitable for capturing the two-dimensional character of the toroidal expanse and crossing number problems, we put forward the following concept improving upon the original "orthogonal width" of [20]. We remark in passing that although our paper does not use nor provide an algorithm to compute the stretch of an embedding, this can be done efficiently on any surface by [6].…”

“…We remark in passing that although our paper does not use nor provide an algorithm to compute the stretch of an embedding, this can be done efficiently on any surface by [6].…”

“…To this end, we define a new density parameter of embedded graphs that captures the truly two-dimensional character of our problem; the stretch of an embedded graph in Definition 2.7. The notion of stretch first appeared in the conference paper [19] and has also been studied from an algorithmic point of view in [6].…”