graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, 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 mor ...

. There are several ways to define series–parallel graphs.

First definition

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 algor ...

. 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 In graph theory, a branch of mathematics, the disjoint union of graphs is an operation that combines two or more graphs to form a larger graph. It is analogous to the disjoint union of sets and is constructed by making the vertex set of the resu ...

''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.

Second 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 ...

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 In graph theory, a biconnected component or block (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. Th ...

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 In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to K₄. 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 explores the relationships between these classifications. A computational problem ...

, because a number of standard graph problems are solvable in linear time on SP-graphs, including finding of the maximum matching, maximum independent set, minimum dominating set and Hamiltonian completion. Some of these problems are

NP-complete In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ...

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. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repea ...

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 ''S'' 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

{{DEFAULTSORT:Series-parallel graph Graph families Graph operations Planar graphs