Steiner trees, partial 2–trees, and minimum IFI networks

Abstract

Minimum isolated failure immune networks are shown to be 2-trees. Further, subgraphs of 2-trees are shown to be exactly those graphs which contain no subgraph homeomorphic to the four-vertex complete graph. Together, these two characterizations yield a linear time algorithm for adding lines to a network to produce a minimum isolated failure immune network, whenever this is possible. This same algorithm, in conjunction with a linear time Steiner tree algorithm for 2-trees, yields a linear time Steiner tree algorithm for partial 2-trees. This contrasts with the known NP-completeness of the Steiner tree problem for planar graphs.

Solving a selected class of location problems by exploiting problem structure: A decomposition approach

“…A way to think about the embedding process is that an elimination sequence is determined, and extra (artificial) edges may have to be added to ensure that the vertices adjacent to y [t ] (when y [t ] is eliminated) are a clique. We note that a partial 2-tree can be embedded in a 2-tree in O(m) (Wald and Colbourn [33]), and that a partial 3-tree can be embedded in a 3-tree in O(m) (Matousek and Thomas [27]). For arbitrary k ú 3, Arnborg, Corneil, and Proskurowski [3] give an O(m k/ 2 ) algorithm for embedding a partial k-tree in a k-tree.…”

Minimum Size Tree-decompositions

“…Very recently, a single-exponential (in k) algorithm has been proposed that computes a tree-decomposition with width at most 5k in the class of graphs with treewidth at most k [4]. As far as we know, the only practical algorithms for computing optimal tree-decompositions hold for graphs with treewidth at most 1 (trivial since tw(G) = 1 if and only if G is a tree), 2 (graphs excluding K 4 as a minor) [13], 3 [2, 9, 10] and 4 [12].…”

m-Median and m-Center Problems with Mutual Communication: Solvable Special Cases

“…(Richey, 1989 To obtain insight into the structural nature of series-parallel graphs it is useful to introduce the concept of a 2-tree. A 2-tree (Wald and Colbourn, 1983;Rardin, Parker, and Richey, 1982) is defined recursively as follows: A triangle is a 2-tree. Given any arc (x,y) of a 2-tree, by appending a node z and adding edges (x,y) and (y,z), the resulting graph is also a 2-tree.…”

“…Given any arc (x,y) of a 2-tree, by appending a node z and adding edges (x,y) and (y,z), the resulting graph is also a 2-tree. The relationship between series-parallel graphs and 2-trees is that (Wald and Colbourn, 1983) Step 1 : Let u and v be the two nodes adjacent to node q in G, where q*u*v.…”