Maintaining Triconnected Components Under Node Expansion

Abstract

SPQR-trees model the decomposition of a graph into triconnected components. In this paper, we study the problem of dynamically maintaining an SPQR-tree while expanding vertices into arbitrary biconnected graphs. This allows us to efficiently merge two SPQR-trees by identifying the edges incident to two vertices with each other. We do this working along an axiomatic definition lifting the SPQR-tree to a stand-alone data structure that can be modified independently from the graph it might have been derived from. Making changes to this structure, we can now observe how the graph represented by the SPQR-tree changes, instead of having to reason which updates to the SPQR-tree are necessary after a change to the represented graph. Using efficient expansions and merges allows us to improve the runtime of the Synchronized Planarity algorithm by Bläsius et al. [2] from $$O(m^2)$$ to $$O(m\cdot \varDelta )$$ , where $$\varDelta $$ is the maximum pipe degree. This also reduces the time for solving several constrained planarity problems, e.g. for Clustered Planarity from $$O((n+d)^2)$$ to $$O(n+d\cdot \varDelta )$$ , where d is the total number of crossings between cluster borders and edges and $$\varDelta $$ is the maximum number of edge crossings on a single cluster border.