Collective additive tree spanners for circle graphs and polygonal graphs

A graph G=(V,E) is said to admit a system of @m collective additive tree r-spanners if there is a system T(G) of at most @m 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 2log"3"2n collective additive tree 2-spanners and every n-vertex k-polygonal graph admits a system of at most 2log"3"2k+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.