Additive Spanners for Circle Graphs and Polygonal Graphs

Abstract

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.

This work was supported by the European Regional Development Fund (ERDF) and by NSERC.