Intersection graphs of paths in a tree
Journal of Combinatorial Theory Series B
Reconstructing the shape of a tree from observed dissimilarity data
Advances in Applied Mathematics
Tree spanners: spanning trees that approximate distances
Tree spanners: spanning trees that approximate distances
SIAM Journal on Discrete Mathematics
Intersection graphs of vertex disjoint paths in a tree
Discrete Mathematics
Tree 3-spanners on interval, permutation and regular bipartite graphs
Information Processing Letters
Restrictions of minimum spanner problems
Information and Computation
Distance approximating trees for chordal and dually chordal graphs
Journal of Algorithms
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
Tree spanners in planar graphs
Discrete Applied Mathematics - Special issue on international workshop of graph-theoretic concepts in computer science WG'98 conference selected papers
The separator theorem for rooted directed vertex graphs
Journal of Combinatorial Theory Series B
Tree spanners on chordal graphs: complexity and algorithms
Theoretical Computer Science
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms
Discrete Applied Mathematics
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A spanning tree T of a graph G is called a treet-spanner, if the distance between any two vertices in T is at most t-times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t=4, and is linearly solvable for t=1 and t=2. The case t=3 still remains open. A directed path graph is called a 2-sep directed path graph if all of its minimal a-b vertex separator for every pair of non-adjacent vertices a and b are of size two. Le and Le [H.-O. Le, V.B. Le, Optimal tree 3-spanners in directed path graphs, Networks 34 (2) (1999) 81-87] showed that directed path graphs admit tree 3-spanners. However, this result has been shown to be incorrect by Panda and Das [B.S. Panda, Anita Das, On tree 3-spanners in directed path graphs, Networks 50 (3) (2007) 203-210]. In fact, this paper observes that even the class of 2-sep directed path graphs, which is a proper subclass of directed path graphs, need not admit tree 3-spanners in general. It, then, presents a structural characterization of tree 3-spanner admissible 2-sep directed path graphs. Based on this characterization, a linear time recognition algorithm for tree 3-spanner admissible 2-sep directed path graphs is presented. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep directed path graph is proposed.