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.
![](../I/Series_parallel_composition.svg.png.webp)
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 by David Eppstein.[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.
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.
Properties
Every series-parallel graph has treewidth at most 2 and branchwidth at most 2.[3] 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.[4][5] 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.
2-connected series-parallel graphs are characterised by having no subgraph homeomorphic to K4.[3]
Series parallel graphs may also be characterized by their ear decompositions.[1]
Computational complexity
SP-graphs may be recognized in linear time[6] 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 SP-graphs,[7] 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.
Generalization
The generalized series-parallel graphs (GSP-graphs) are an extension of the SP-graphs[8] 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 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, which are analogous to the series composition operations in series-parallel graphs, P-nodes, which are analogous to the parallel composition operations in series-parallel graphs, and R-nodes, which 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.
References
- Eppstein, David (1992). "Parallel recognition of series-parallel graphs" (PDF). Information and Computation. 98 (1): 41–55. doi:10.1016/0890-5401(92)90041-D.
- 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.
- Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy P. (1999). Graph classes: a survey. SIAM Monographs on Discrete Mathematics. and Applications. 3. Philadelphia, PA: Society for Industrial and Applied Mathematics. pp. 172–174. ISBN 978-0-898714-32-6. Zbl 0919.05001.
- 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. hdl:1874/18312.
- 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. S2CID 16082154.
- 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 (in Russian)