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.
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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 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.
[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] Properties
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.[3][4] 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. Graphs of treewidth at most 2 have an explicit forbidden minor characterization, implying that a graph is series-parallel if and only if its biconnected components are linked in a path and it excludes the complete graph K4 as a minor.
Series parallel graphs may also be characterized by their ear decompositions.[1]
[edit] Research involving series-parallel graphs
SPGs may be recognized in linear time[5] 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 graph problems are solvable in linear time on SPGs,[6] 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[7] 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.
An SPQR tree is a tree structure that can be defined for an arbitrary 2-vertex-connected graph. It has S nodes that are analogous to the series composition operations in series-parallel graphs, P nodes that are analogous to the parallel composition operations in series-parallel graphs, and R nodes that do not correspond to series-parallel composition operations. A 2-connected graph is series-parallel if and only if there are no R nodes in its SPQR tree.
[edit] See also
[edit] References
- ^ a b Eppstein, David (1992). "Parallel recognition of series-parallel graphs". Information and Computation 98 (1): 41–55. doi:10.1016/0890-5401(92)90041-D. http://www.ics.uci.edu/~eppstein/pubs/Epp-IC-92.pdf.
- ^ Duffin, R. J. (1965). "Topology of Series-Parallel Networks". Journal of Mathematical Analysis and Applications 10 (2): 303–313. doi:10.1016/0022-247X(65)90125-3.
- ^ Bodlaender, H. (1998). "A partial k-arboretum of graphs with bounded treewidth". Theoretical Computer Science 209 (1–2): 1–45. doi:10.1016/S0304-3975(97)00228-4.
- ^ Hall, Rhiannon; Oxley, James; Semple, Charles; Whittle, Geoff (2002). "On matroids of branch-width three". Journal of Combinatorial Theory, Series B 86 (1): 148–171. doi:10.1006/jctb.2002.2120.
- ^ Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982). "The recognition of series parallel digraphs". SIAM Journal on Computing 11 (2): 289–313. doi:10.1137/0211023.
- ^ Takamizawa, K.; Nishizeki, T.; Saito, N. (1982). "Linear-time computability of combinatorial problems on series-parallel graphs". Journal of the ACM 29 (3): 623–641. doi:10.1145/322326.322328.
- ^ Korneyenko, N. M. (1994). "Combinatorial algorithms on a class of graphs". Discrete Applied Mathematics 54 (2–3): 215–217. doi:10.1016/0166-218X(94)90022-1. Translated from Notices of the BSSR Academy of Sciences, Ser. Phys.-Math. Sci., (1984) no. 3, pp. 109-111 (Russian)
