Mondshein Sequences (a.k.a. (2,1)-Orders)

Abstract

Canonical orderings have been used as a key tool in graph drawing, graph encoding, and visibility representations for the last decades [H. de Fraysseix, J. Pach, and R. Pollack, Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC '88), ACM, New York, 1988, pp. 426--433; G. Kant, Proceedings of the 33rd Annual Symposium on Foundations of Computer Science (FOCS '92), IEEE Press, Piscataway, NJ, 1992, pp. 101--110]. We study a far-reaching generalization of canonical orderings to nonplanar graphs that was published by Lee Mondshein in a Ph.D. thesis as early as 1971. Mondshein proposed to order the vertices of a graph in a sequence such that for any $i$, the vertices from 1 to $i$ essentially induce a 2-connected graph, while the remaining vertices from $i+1$ to $n$ induce a connected graph. Mondshein's sequence generalizes canonical orderings and later became independently known as nonseparating ear decomposition. Surprisingly, this fundamental link between canonical orderings and nons...