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

Collective Tree Spanners in Graphs with Bounded Genus, Chordality, Tree-Width, or Clique-Width

Abstract. In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V,E) admits a system of µ collective additive tree r-spanners if there is asystemT (G) ofatmostµ spanning trees of G such that for any two vertices x,y of G a spanning tree T ∈T(G) exists such that dT(x,y) ≤ dG(x,y)+r. We describe a general method for constructing a ”small ” system of collective additive tree r-spanners with small values of r for ”well” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O ( √ n) collective additive tree 0–spanners, any weighted graph with tree-width at most k −1 admits a system of k log 2 n collective additive tree 0–spanners, any weighted graph with clique-width at most k admits a system of k log 3/2 n collective additive tree (2w)–spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log 2 n collective additive tree (2⌊c/2⌋w)–spanners and a system of 4log 2 n collective additive tree (2(⌊c/3⌋+1)w)–spanners (here, w is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4log 2 n collective additive tree (2w)-spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum