Tree spanners of bounded degree graphs

A tree $t$-spanner of a graph $G$ is a spanning tree of $G$ such that the distance between pairs of vertices in the tree is at most $t$ times their distance in $G$. Deciding tree $t$-spanner admissible graphs has been proved to be tractable for $t3$, while the complexity status of this problem is unresolved when $t=3$. For every $t>2$ and $b>0$, an efficient dynamic programming algorithm to decide tree $t$-spanner admissibility of graphs with vertex degrees less than $b$ is presented. Only for $t=3$, the algorithm remains efficient, when graphs $G$ with degrees less than $b\log |V(G)|$ are examined

Collective Additive Tree Spanners of Bounded Tree-Breadth Graphs with Generalizations and Consequences

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph $G$ {\em admits a system of $\mu$ collective additive tree $r$-spanners} (resp., {\em multiplicative tree $t$-spanners}) if there is a system \cT(G) of at most $\mu$ spanning trees of $G$ such that for any two vertices $x,y$ of $G$ a spanning tree T\in \cT(G) exists such that $d_T(x,y)\leq d_G(x,y)+r$ (resp., $d_T(x,y)\leq t\cdot d_G(x,y)$). When $\mu=1$ one gets the notion of {\em additive tree $r$-spanner} (resp., {\em multiplicative tree $t$-spanner}). It is known that if a graph $G$ has a multiplicative tree $t$-spanner, then $G$ admits a Robertson-Seymour's tree-decomposition with bags of radius at most $\lceil{t/2}\rceil$ in $G$. We use this to demonstrate that there is a polynomial time algorithm that, given an $n$-vertex graph $G$ admitting a multiplicative tree $t$-spanner, constructs a system of at most $\log_2 n$ collective additive tree $O(t\log n)$-spanners of $G$. That is, with a slight increase in the number of trees and in the stretch, one can "turn" a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph $G$ admits a multiplicative $t$-spanner with tree-width $k-1$, then $G$ admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most $k$ disks of $G$ of radius at most $\lceil{t/2}\rceil$ each. This is used to demonstrate that, for every fixed $k$, there is a polynomial time algorithm that, given an $n$-vertex graph $G$ admitting a multiplicative $t$-spanner with tree-width $k-1$, constructs a system of at most $k(1+ \log_2 n)$ collective additive tree $O(t\log n)$-spanners of $G$.Comment: 21 pages, 4 figure