Tree t-Spanners of a Graph: Minimizing Maximum Distances Efficiently
A tree t-spanner of a graph G is a spanning subtree T in which the distance between any two adjacent vertices of G is at most t. The smallest t for which G has a tree t-spanner is the tree stretch index. The problem of determining the tree stretch index has been studied by: establishing lower and upper bounds, based, for instance, on the girth value and on the minimum diameter spanning tree problem, respectively; and presenting some classes for which t is a tight value. Moreover, in 1995, the computational complexities of determining whether \(t = 2\) or \(t \ge 4\) were settled to be polynomially time solvable and NP-complete, respectively, while deciding if \(t = 3\) still remains an open problem.
With respect to the computational complexity aspect of this problem, we present an inconsistence on the sufficient condition of tree 2-spanner admissible graphs. Moreover, while dealing with operations in graphs, we provide optimum tree t-spanners for 2 cycle-power graphs and for prism graphs, which are obtained from 2 cycle-power graphs after removing a perfect matching. Specifically, the stretch indexes for both classes are far from their girth’s natural lower bounds, and surprisingly, the parameter does not change after such a matching removal. We also present efficient strategies to obtain optimum tree t-spanners considering threshold graphs, split graphs, and generalized octahedral graphs. With this last result in addition to vertices addition operations and the tree decomposition of a cograph, we are able to present the stretch index for cographs.
KeywordsTree t-spanner Stretch index Lower bounds Generalized octahedral graph Cycle-power graph Prism graph Threshold graph Split graph Cograph
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