Abstract
A tree t-spanner of a graph G is a spanning tree T of G in which any two adjacent vertices of G have distance at most t in T. The line graph L(G) of a graph G is the intersection graph of the edges of G. We define the edge tree t-spanner of a graph G as a spanning tree T of L(G) in which any two edges that share an endpoint in G have distance at most t in T. Although determining if G has a tree 3-spanner is an open problem for more than 20 years, we settle that deciding if a graph G has an edge tree 3-spanner is polynomial-time solvable. As a consequence, we present polynomial time algorithms for the edge tree t-spanner problem for several graph classes such as trees, join of graphs, split graphs, P 4-tidy, and (1, 2)-graphs. Moreover, we establish that deciding whether a graph G has an edge tree 8-spanner is NP-complete, even if G is bipartite.
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This study was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.
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Couto, F., Cunha, L., Posner, D. (2021). Edge Tree Spanners. In: Gentile, C., Stecca, G., Ventura, P. (eds) Graphs and Combinatorial Optimization: from Theory to Applications. AIRO Springer Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-63072-0_16
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DOI: https://doi.org/10.1007/978-3-030-63072-0_16
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