Worst-Case Polylog Incremental SPQR-trees: Embeddings, Planarity, and Triconnectivity

Fully-dynamic Planarity Testing in Polylogarithmic Time

“…If a planar graph has no vertex cut-sets of size ≤ 2, its embedding is unique up to reflection. On the other hand, if a plane graph has an articulation point (cut vertex) or a separation pair (2-vertex cut), then it may be possible to alter the embedding by flipping [21,11,12] the embedding in that point or pair (see figure 1). Given two embeddings of the same graph, the flip-distance between them is the minimal number of flips necessary to get from one to the other.…”

“…The initial cutting and the final gluing involve the same vertices but not necessarily the same faces, thus, the graph but not the embedding is preserved. Local changes to the embedding of a graph [12].…”

“…The (skeleton graphs associated with) the B nodes are sometimes referred to as G's biconnected components. In this paper, we use the term relaxed BC tree as defined in [12] to denote a tree that satisfies all but the last condition. Unlike the strict BC tree, the relaxed BC tree is not unique.…”

“…[5]). In this paper, we use the term relaxed SPQR tree as defined in [12] to denote a tree that satisfies all but the last condition. Unlike the strict SPQR tree, the relaxed SPQR tree is not unique.…”

“…local changes to the embedding, and, so that it may handle edge-insertions that only require one such flip. In [12], we analyze these flips further and show that there exists a class of embeddings where only Θ(log n) flips are needed to accommodate any one edge insertion that preserves planarity.…”