Simpler and efficient characterizations of tree t-spanners for graphs with few P4's and (k, l)-graphs
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A tree t-spanner of a graph G is a spanning tree 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 called tree stretch index. The t-admissibility problem aims to decide whether the tree stretch index is at most t. Regarding its optimization version, the smallest t for which G is t-admissible is the stretch index of G, denoted by σ_T(G). Given a graph with n vertices and m edges, the recognition of 2-admissible graphs can be done O(n+m) time, whereas t-admissibility is NP-complete for σ_T(G) ≤ t, t ≥ 4 and deciding if t = 3 is an open problem, for more than 20 years. Since the structural knowledge of classes can be determinant to classify 3-admissibility's complexity, in this paper we present simpler and faster algorithms to check 2 and 3-admissibility for families of graphs with few P_4's and (k,ℓ)-graphs. Regarding (0,ℓ)-graphs, we present lower and upper bounds for the stretch index of these graphs and characterize graphs whose stretch indexes are equal to the proposed upper bound. Moreover, we prove that t-admissibility is NP-complete even for line graphs of subdivided graphs.
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