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SPQR tree
In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An SPQR tree is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time and has several applications in dynamic graph algorithms and graph drawing.
The basic structures underlying the SPQR tree, the triconnected components of a graph, and the connection between this decomposition and the planar embeddings of a planar graph, were first investigated by Saunders Mac Lane (1937); these structures were used in efficient algorithms by several other researchers prior to their formalization as the SPQR tree by Di Battista and Tamassia (1989, 1990, 1996).
An SPQR tree takes the form of an unrooted tree in which for each node x there is associated an undirected graph or multigraph Gx. The node, and the graph associated with it, may have one of four types, given the initials SPQR:
Each edge xy between two nodes of the SPQR tree is associated with two directed virtual edges, one of which is an edge in Gx and the other of which is an edge in Gy. Each edge in a graph Gx may be a virtual edge for at most one SPQR tree edge.
An SPQR tree T represents a 2-connected graph GT, formed as follows. Whenever SPQR tree edge xy associates the virtual edge ab of Gx with the virtual edge cd of Gy, form a single larger graph by merging a and c into a single supervertex, merging b and d into another single supervertex, and deleting the two virtual edges. That is, the larger graph is the 2-clique-sum of Gx and Gy. Performing this gluing step on each edge of the SPQR tree produces the graph GT; the order of performing the gluing steps does not affect the result. Each vertex in one of the graphs Gx may be associated in this way with a unique vertex in GT, the supervertex into which it was merged.
Typically, it is not allowed within an SPQR tree for two S nodes to be adjacent, nor for two P nodes to be adjacent, because if such an adjacency occurred the two nodes could be merged into a single larger node. With this assumption, the SPQR tree is uniquely determined from its graph. When a graph G is represented by an SPQR tree with no adjacent P nodes and no adjacent S nodes, then the graphs Gx associated with the nodes of the SPQR tree are known as the triconnected components of G.
The SPQR tree of a given 2-vertex-connected graph can be constructed in linear time.
The problem of constructing the triconnected components of a graph was first solved in linear time by Hopcroft & Tarjan (1973). Based on this algorithm, Di Battista & Tamassia (1996) suggested that the full SPQR tree structure, and not just the list of components, should be constructible in linear time. After an implementation of a slower algorithm for SPQR trees was provided as part of the GDToolkit library, Gutwenger & Mutzel (2001) provided the first linear-time implementation. As part of this process of implementing this algorithm, they also corrected some errors in the earlier work of Hopcroft & Tarjan (1973).
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SPQR tree AI simulator
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SPQR tree
In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An SPQR tree is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time and has several applications in dynamic graph algorithms and graph drawing.
The basic structures underlying the SPQR tree, the triconnected components of a graph, and the connection between this decomposition and the planar embeddings of a planar graph, were first investigated by Saunders Mac Lane (1937); these structures were used in efficient algorithms by several other researchers prior to their formalization as the SPQR tree by Di Battista and Tamassia (1989, 1990, 1996).
An SPQR tree takes the form of an unrooted tree in which for each node x there is associated an undirected graph or multigraph Gx. The node, and the graph associated with it, may have one of four types, given the initials SPQR:
Each edge xy between two nodes of the SPQR tree is associated with two directed virtual edges, one of which is an edge in Gx and the other of which is an edge in Gy. Each edge in a graph Gx may be a virtual edge for at most one SPQR tree edge.
An SPQR tree T represents a 2-connected graph GT, formed as follows. Whenever SPQR tree edge xy associates the virtual edge ab of Gx with the virtual edge cd of Gy, form a single larger graph by merging a and c into a single supervertex, merging b and d into another single supervertex, and deleting the two virtual edges. That is, the larger graph is the 2-clique-sum of Gx and Gy. Performing this gluing step on each edge of the SPQR tree produces the graph GT; the order of performing the gluing steps does not affect the result. Each vertex in one of the graphs Gx may be associated in this way with a unique vertex in GT, the supervertex into which it was merged.
Typically, it is not allowed within an SPQR tree for two S nodes to be adjacent, nor for two P nodes to be adjacent, because if such an adjacency occurred the two nodes could be merged into a single larger node. With this assumption, the SPQR tree is uniquely determined from its graph. When a graph G is represented by an SPQR tree with no adjacent P nodes and no adjacent S nodes, then the graphs Gx associated with the nodes of the SPQR tree are known as the triconnected components of G.
The SPQR tree of a given 2-vertex-connected graph can be constructed in linear time.
The problem of constructing the triconnected components of a graph was first solved in linear time by Hopcroft & Tarjan (1973). Based on this algorithm, Di Battista & Tamassia (1996) suggested that the full SPQR tree structure, and not just the list of components, should be constructible in linear time. After an implementation of a slower algorithm for SPQR trees was provided as part of the GDToolkit library, Gutwenger & Mutzel (2001) provided the first linear-time implementation. As part of this process of implementing this algorithm, they also corrected some errors in the earlier work of Hopcroft & Tarjan (1973).