Dynamic planar embeddings of dynamic graphs

TY - GEN

T1 - Dynamic planar embeddings of dynamic graphs

AU - Holm, Jacob

AU - Rotenberg, Eva

N1 - Conference code: 32

PY - 2015/2/1

Y1 - 2015/2/1

N2 - 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(log2 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.

AB - 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(log2 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.

U2 - 10.4230/LIPIcs.STACS.2015.434

DO - 10.4230/LIPIcs.STACS.2015.434

M3 - Article in proceedings

SP - 434

EP - 446

BT - 32nd Symposium on Theoretical Aspects of Computer Science (STACS 2015)

A2 - Mayr, Ernst W.

A2 - Ollinger, Nicolas

PB - Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

T2 - 32nd International Symposium on Theoretical Aspects of Computer Science

Y2 - 4 March 2015 through 7 March 2015

ER -