Additive Spanners for Circle Graphs and Polygonal Graphs

A graph G = (V ,E ) is said to admit a system of μ collective additive tree r -spanners if there is a system $\cal{T}$(G ) of at most μ spanning trees of G such that for any two vertices u ,v of G a spanning tree $T\in \cal{T}$(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 $2\log_{\frac{3}{2}}n$ collective additive tree 2-spanners and every n -vertex k -polygonal graph admits a system of at most $2\log_{\frac{3}{2}}k+7$ 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 3 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.