Minimum t-Spanners on Subcubic Graphs

Abstract

For a constant $t \ge 1$, a t-spanner of a connected graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most t times its distance in G. We address two problems on spanners: the minimum t-spanner problem (MinS $_t$), and a minimization version of the tree t-spanner problem (TreeS $_t$). MinS $_t$ seeks a t-spanner with minimum number of edges. TreeS $_t$ is a decision problem concerning the existence of a t-spanner that is a tree. The concept of spanner was introduced by Peleg & Ullman in 1989, in a context regarding the construction of optimal synchronizers for the hypercube. MinS $_t$ is known to be $\textsc {NP}$-hard for every $t \ge 2$ even on some bounded-degree graphs. TreeS $_t$ is polynomially solvable for $t=2$ and $\textsc {NP}$-complete for $t \ge 4$, but its complexity for $t=3$ remains open.

We investigate both MinS $_3$ and TreeS $_2$ on the class of subcubic graphs. We prove that MinS $_3$ can be solved in polynomial time, using a similar technique as the one used by Cai & Keil (1994) for $t=2$. This result also gives an alternative algorithm to solve TreeS $_3$ in polynomial time. Additionally, we study TreeS $_2$ from a polyhedral point-of-view and show a complete linear characterization of the associated polytope. This result, interesting on its own right, gives a polynomial-time algorithm to solve a natural minimization version of TreeS $_2$ on subcubic graphs with costs assigned to its edges.