Series-parallel graph
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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.
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[edit] Definition and terminology
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 in[1]
A two-terminal graph (TTG) is a graph with two distinguished vertices, s and t called source and sink, respectively.
The parallel composition P = P(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 P and merging the sinks of X and Y to create the sink of P.
The series composition S = S(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 source of P and the sink of Y becomes the sink of P.
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.
[edit] Alternative definition
The following definition specifies the same class of graphs[2].
Definition 2. A graph is an sp-graph, if it may be turned into K2 by a sequence of the following operations:
- Replacement of a pair of parallel edges with a single edge that connects their common endpoints
- Replacement of a pair of edges incident to a vertex of degree 2 other than s or t with a single edge.
[edit] Research involving series-parallel graphs
SPGs may be recognized in linear time[3] and their series-parallel decomposition may be constructed in linear time as well.
Besides being a model of certain types of electric networks, these graphs are of interest in computational complexity theory, because a number of standard problems on graphs are solvable in linear time,[4] including finding of the maximum matching, maximum independent set, minimum dominating set and Hamiltonian completion. Some of these problems are NP-complete for general graphs. The solution capitalizes on the fact that if the answers for one of these problems are known for two SP-graphs, then one can quickly find the answer for their series and parallel compositions.
The series-parallel networks problem refers to a graph enumeration problem which asks for the number of series-parallel graphs that can be formed using a given number of edges.
[edit] Generalization
The generalized series-parallel graphs (GSP-graphs) are an extension of the SPGs[5] with the same algorithmic efficiency for the mentioned problems. The class of GSP-graphs include the classes of SP-graphs and outerplanar graphs.
GSP graphs may be specified by the Definition 2 augmented with the third operation of deletion of a dangling vertex (vertex of degree 1). Alternatively, Definition 1 may be augmented with the following operation.
- The source merge S = M(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the source of X with the source of Y. The source and sink of X become the source and sink of P respectively.
[edit] References
- ^ Eppstein, David (1992). "Parallel recognition of series-parallel graphs". Information and Computation 98 (1): 41–55. doi:.
- ^ Duffin, R. J. (1965). "Topology of Series-Parallel Networks". Journal of Mathematical Analysis and Applications 10: 303–313. doi:.
- ^ Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982). "The recognition of series parallel digraphs". SIAM Journal on Computing 11: 289–313. doi:.
- ^ Takamizawa, K.; Nishizeki, T.; Saito, N. (1982). "Linear-time computability of combinatorial problems on series-parallel graphs". Journal of the ACM 29: 623–641. doi:.
- ^ Korneyenko, N. M. (1994). "Combinatorial algorithms on a class of graphs". Discrete Applied Mathematics 54 (2–3): 215–217. doi:. Translated from Notices of the BSSR Academy of Sciences, Ser. Phys.-Math. Sci., (1984) no. 3, pp. 109-111 (Russian)
