Collective additive tree spanners for circle graphs and polygonal graphs

AbstractA graph G=(V,E) is said to admit a system of μ collective additive tree r-spanners if there is a system T(G) of at most μ spanning trees of G such that for any two vertices u,v of G a spanning tree T∈T(G) exists such that the distance in T between u and v is at most r plus their distance in G. In this paper, we examine the problem of finding “small” systems of collective additive tree r-spanners for small values of r on circle graphs and on polygonal graphs. Among other results, we show that every n-vertex circle graph admits a system of at most 2log32n collective additive tree 2-spanners and every n-vertex k-polygonal graph admits a system of at most 2log32k+7 collective additive tree 2-spanners. Moreover, we show that every n-vertex k-polygonal graph admits an additive (k+6)-spanner with at most 6n−6 edges and every n-vertex 3-polygonal graph admits a system of at most three collective additive tree 2-spanners and an additive tree 6-spanner. All our collective tree spanners as well as all sparse spanners are constructible in polynomial time

Additive sparse spanners for graphs with bounded length of largest induced cycle

AbstractIn this paper, we show that every chordal graph with n vertices and m edges admits an additive 4-spanner with at most 2n-2 edges and an additive 3-spanner with at most O(nlogn) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory 13 (1989) 99–116]. Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner is constructable in O(mlogn) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k+1)-spanner with at most 2n-2 edges which is constructable in O(nk+m) time