In
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, 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.
Definition and terminology
In this context, the term graph means
multigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by mo ...
.
There are several ways to define series–parallel graphs. The following definition basically follows the one used by
David Eppstein
David Arthur Eppstein (born 1963) is an American computer scientist and mathematician. He is a Distinguished Professor of computer science at the University of California, Irvine. He is known for his work in computational geometry, graph algo ...
.
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.
''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
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests. The gr ...
at most 2 and
branchwidth
In graph theory, a branch-decomposition of an undirected graph ''G'' is a hierarchical clustering of the edges of ''G'', represented by an unrooted binary tree ''T'' with the edges of ''G'' as its leaves. Removing any edge from ''T'' partitions ...
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
In graph theory, a biconnected component (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph. The blocks ...
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.
2-connected series–parallel graphs are characterised by having no subgraph
homeomorphic to K
4.
[
Series parallel graphs may also be characterized by their ear decompositions.][
]
Computational complexity
SP-graphs may be recognized in linear time 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
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...
, because a number of standard graph problems are solvable in linear time on SP-graphs, including finding of the maximum matching
Maximum cardinality matching is a fundamental problem in graph theory.
We are given a graph , and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is ad ...
, maximum independent set, minimum dominating set and Hamiltonian completion. Some of these problems are NP-complete
In computational complexity theory, a problem is NP-complete when:
# it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by tryin ...
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[ Translated from ''Notices of the BSSR Academy of Sciences, Ser. Phys.-Math. Sci.'', (1984) no. 3, pp. 109–111 ] with the same algorithmic efficiency
In computer science, algorithmic efficiency is a property of an algorithm which relates to the amount of computational resources used by the algorithm. An algorithm must be analyzed to determine its resource usage, and the efficiency of an al ...
for the mentioned problems. The class of GSP-graphs include the classes of SP-graphs and outerplanar graph
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.
Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two fo ...
s.
GSP graphs may be specified by ''Definition 2'' augmented with the third operation of deletion
Deletion or delete may refer to:
Computing
* File deletion, a way of removing a file from a computer's file system
* Code cleanup, a way of removing unnecessary variables, data structures, cookies, and temporary files in a programming language
* ...
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.
See also
* Threshold graph
* Cograph
In graph theory, a cograph, or complement-reducible graph, or ''P''4-free graph, is a graph that can be generated from the single-vertex graph ''K''1 by complementation and disjoint union. That is, the family of cographs is the smallest class of ...
* Hanner polytope
* Series-parallel partial order
References
{{DEFAULTSORT:Series-parallel graph
Graph families
Graph operations
Planar graphs