10 research outputs found
Dynamic Planar Embeddings of Dynamic Graphs
We present an algorithm to support the dynamic embedding in the plane of a
dynamic graph. An edge can be inserted across a face between two vertices on
the face boundary (we call such a vertex pair linkable), and edges can be
deleted. The planar embedding can also be changed locally by flipping
components that are connected to the rest of the graph by at most two vertices.
Given vertices , linkable decides whether and are
linkable in the current embedding, and if so, returns a list of suggestions for
the placement of in the embedding. For non-linkable vertices , we
define a new query, one-flip-linkable providing a suggestion for a flip
that will make them linkable if one exists. We support all updates and queries
in O(log) time. Our time bounds match those of Italiano et al. for a
static (flipless) embedding of a dynamic graph.
Our new algorithm is simpler, exploiting that the complement of a spanning
tree of a connected plane graph is a spanning tree of the dual graph. The
primal and dual trees are interpreted as having the same Euler tour, and a main
idea of the new algorithm is an elegant interaction between top trees over the
two trees via their common Euler tour.Comment: Announced at STACS'1
Dynamic Planar Embeddings of Dynamic Graphs
We present an algorithm to support the dynamic embedding in the plane
of a dynamic graph. An edge can be inserted across a face between two vertices on the boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Given vertices u,v, linkable(u,v) decides whether u and v are linkable, and if so, returns a list of suggestions for the placement of (u,v) in the embedding. For non-linkable vertices u,v, we define a new query, one-flip-linkable(u,v) providing a suggestion for a flip that will make them linkable if one exists. We will support all updates and queries in O(log^2 n) time. Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph.
Our new algorithm is simpler, exploiting that the complement of a spanning tree of a connected plane graph is a spanning tree of the dual graph. The primal and dual trees are interpreted as having the same Euler tour, and a main idea of the new algorithm is an elegant
interaction between top trees over the two trees via their common Euler tour
Fully-dynamic Planarity Testing in Polylogarithmic Time
Given a dynamic graph subject to insertions and deletions of edges, a natural
question is whether the graph presently admits a planar embedding. We give a
deterministic fully-dynamic algorithm for general graphs, running in amortized
time per edge insertion or deletion, that maintains a bit
indicating whether or not the graph is presently planar. This is an exponential
improvement over the previous best algorithm [Eppstein, Galil, Italiano,
Spencer, 1996] which spends amortized time per update.Comment: Updated version of paper submitted to STOC'20. This version features
a complete rewrite of section 4.4 (do-separation-flips). The new version
fixes an overlooked case in the previous version (the two fundamental cycles
we find do not necessarily share an edge) and contains a detailed
case-by-case proof of correctnes
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum