Series-parallel graph
In graph theory, series-parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits.
In this context, the term graph means multigraph.
There are several ways to define series-parallel graphs. The following definition basically follows the one used by David Eppstein.
A two-terminal graph (TTG) is a graph with two distinguished vertices, s and t called source and sink, respectively.
The parallel composition Pc = Pc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sources of X and Y to create the source of Pc and merging the sinks of X and Y to create the sink of Pc.
The series composition Sc = Sc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sink of X with the source of Y. The source of X becomes the source of Sc and the sink of Y becomes the sink of Sc.
A two-terminal series-parallel graph (TTSPG) is a graph that may be constructed by a sequence of series and parallel compositions starting from a set of copies of a single-edge graph K2 with assigned terminals.
Definition 1. Finally, a graph is called series-parallel (sp-graph), if it is a TTSPG when some two of its vertices are regarded as source and sink.
In a similar way one may define series-parallel digraphs, constructed from copies of single-arc graphs, with arcs directed from the source to the sink.
The following definition specifies the same class of graphs.
Definition 2. A graph is an sp-graph, if it may be turned into K2 by a sequence of the following operations:
Every series-parallel graph has treewidth at most 2 and branchwidth at most 2. Indeed, a graph has treewidth at most 2 if and only if it has branchwidth at most 2, if and only if every biconnected component is a series-parallel graph. The maximal series-parallel graphs, graphs to which no additional edges can be added without destroying their series-parallel structure, are exactly the 2-trees.
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