A tree $t$-spanner $T$ of a simple graph $G$ is a spanning tree of $G$, such that for every pair of vertices of $G$ their distance in $T$ is at most $t$ times their distance in $G$, where $t$ is called a stretch factor of $T$ in $G$. It has been shown that there is a linear time algorithm to find a tree 2-spanner in a graph; it has also been proved that, for each $t>3$, determining whether a graph admits a tree $t$-spanner is an NP-complete problem. This thesis studies tree $t$-spanners from both theoretical and algorithmic perspectives. In particular, it is proved that a nontree graph admits a unique tree $t$-spanner for at most one value of stretch factor $t$. As a corollary, a nontree bipartite graph cannot admit a unique tree $t$-spanner for any $t$. But, for each $t$, there are infinitely many nontree graphs that admit exactly one tree $t$-spanner. Furthermore, for each $t$, let U($t$) be the set of graphs being the union of two tree $t$-spanners of a graph. Although graphs in U(2) do not have cycles of length greater than 4, graphs in U(3) may contain cycles of arbitrary length. It turns out that any even cycle is an induced subgraph of a graph in U(3), while no graph in U(3) contains an induced odd cycle other than a triangle; graphs in U(3) are shown to be perfect. Also, properties of induced even cycles of graphs in U(3) are presented. For each $t>3$, though, graphs in U($t$) may contain induced odd cycles of any length. Moreover, there is an efficient algorithm to recognize graphs that admit a tree 3-spanner of diameter at most 4, while it is proved that, for each $t>3$, determining whether a graph admits a tree $t$-spanner of diameter at most $t+1$ is an NP-complete problem. It is not known if it is hard to recognize graphs that admit a tree 3-spanner of general diameter; however integer programming is employed to provide certificates of tree 3-spanner inadmissibility for a family of graphs.

A tree $t$-spanner $T$ of a simple graph $G$ is a spanning tree of $G$, such that for every pair of vertices of $G$ their distance in $T$ is at most $t$ times their distance in $G$, where $t$ is called a stretch factor of $T$ in $G$. It has been shown that there is a linear time algorithm to find a tree 2-spanner in a graph; it has also been proved that, for each $t>3$, determining whether a graph admits a tree $t$-spanner is an NP-complete problem. This thesis studies tree $t$-spanners from both theoretical and algorithmic perspectives. In particular, it is proved that a nontree graph admits a unique tree $t$-spanner for at most one value of stretch factor $t$. As a corollary, a nontree bipartite graph cannot admit a unique tree $t$-spanner for any $t$. But, for each $t$, there are infinitely many nontree graphs that admit exactly one tree $t$-spanner. Furthermore, for each $t$, let U($t$) be the set of graphs being the union of two tree $t$-spanners of a graph. Although graphs in U(2) do not have cycles of length greater than 4, graphs in U(3) may contain cycles of arbitrary length. It turns out that any even cycle is an induced subgraph of a graph in U(3), while no graph in U(3) contains an induced odd cycle other than a triangle; graphs in U(3) are shown to be perfect. Also, properties of induced even cycles of graphs in U(3) are presented. For each $t>3$, though, graphs in U($t$) may contain induced odd cycles of any length. Moreover, there is an efficient algorithm to recognize graphs that admit a tree 3-spanner of diameter at most 4, while it is proved that, for each $t>3$, determining whether a graph admits a tree $t$-spanner of diameter at most $t+1$ is an NP-complete problem. It is not known if it is hard to recognize graphs that admit a tree 3-spanner of general diameter; however integer programming is employed to provide certificates of tree 3-spanner inadmissibility for a family of graphs.