Tree 3-spanners of diameter at most 5
Ioannis Papoutsakis

TL;DR
This paper presents an efficient algorithm to determine whether a graph admits a tree 3-spanner of diameter at most 5, addressing a previously unresolved problem in graph theory.
Contribution
The paper introduces a novel polynomial-time algorithm for identifying graphs with a tree 3-spanner of diameter at most 5, expanding understanding of spanner diameter constraints.
Findings
Algorithm efficiently decides tree 3-spanner of diameter at most 5.
Graphs with diameter ≤ 3 that admit a tree 3-spanner also admit one of diameter ≤ 5.
The method solves an open problem for certain classes of graphs.
Abstract
Tree spanners approximate distances within graphs; a subtree of a graph is a tree -spanner of the graph if and only if for every pair of vertices their distance in the subtree is at most times their distance in the graph. When a graph contains a subtree of diameter at most , then trivially admits a tree -spanner. Now, determining whether a graph admits a tree -spanner of diameter at most is an NP complete problem, when , and it is tractable, when . Although it is not known whether it is tractable to decide graphs that admit a tree 3-spanner of any diameter, an efficient algorithm to determine graphs that admit a tree 3-spanner of diameter at most 5 is presented. Moreover, it is proved that if a graph of diameter at most 3 admits a tee 3-spanner, then it admits a tree 3-spanner of diameter at most 5. Hence, this algorithm decides tree 3-spanner…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Complexity and Algorithms in Graphs
