Abstract
In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in AT-free graphs. We say that a graph G=(V,E) admits a system of μcollective 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 x,y of G a spanning tree $T\in {\cal T}(G)$ exists such that dT(x,y)≤ dG(x,y)+r. Among other results, we show that AT-free graphs have a system of two collective additive tree 2-spanners (whereas there are trapezoid graphs that do not admit any additive tree 2-spanner). Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those graphs. Also, any DSP-graph (there exists a dominating shortest path) admits one additive tree 4-spanner, a system of two collective additive tree 3-spanners and a system of five collective additive tree 2-spanners.
Highlights
Given a graph G = (V, E), a spanning subgraph H is called a spanner if H provides a “good” approximation of the distances in G
After introducing the notation and definitions used throughout the paper, we examine various families of graphs related to asteroidal triple-free (AT-free) graphs from the perspective of determining whether they have a small constant number of collective additive tree r-spanners for small constant r
For families that strictly contain AT-free graphs, we prove that any DSP-graph admits one additive tree 4-spanner, a system of two collective additive tree 3-spanners and a system of five collectible additive tree 2-spanners
Summary
We say that a graph G = (V, E) admits 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 x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r (a multiplicative variant of this notion can be defined analogously). Note that any graph on n vertices admits a system of at most n − 1 collective additive tree 0-spanners (take n − 1 Breadth-First-Search–trees rooted at different vertices of G). Once one has determined a system of collective additive tree spanners, it is interesting to see how such a system can be used to design compact and efficient routing schemes for the given graph. The quality of a routing scheme is measured in terms of its additive stretch, called deviation, (or multiplicative stretch, called delay), namely, the maximum surplus (or ratio) between the length of a route, produced by the scheme for some pair of vertices, and their distance
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