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. 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.
Read more about this topic: Series-parallel Graph
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