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A tree t-spanner T of a graph G is a spanning tree in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph theoretic, algorithmic and complexity issues about tree spanners. It is shown that a tree 1-spanner, if it exists, in a weighted graph with m edges and n vertices is a minimum spanning tree and can be found in O(m log β(m, n)) time, where β(m, n) = min{i| log (i) n ≤ m/n}. On the other hand, for any fixed t > 1, the problem of determining the existence of a tree t-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree t-spanner is NP-complete for any fixed t ≥ 4. A theorem which captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an O((m + n)α(m, n)) algorithm is provided for finding a tree t-spanner with t as small as possible, where α(m, n) is a functional inverse of Ackerman's function. The results for tree spanners on undirected graphs are extended to "quasitree spanners" on digraphs. Furthermore, linear time algorithms are derived for verifying tree spanners and quasitree spanners.

We develop a new randomized method, random separation, for solving fixed-cardinality optimization problems on graphs, i.e., problems concerning solutions with exactly a fixed number k of elements (e.g., k vertices V) that optimize solution values (e.g., the number of edges covered by V). The key idea of the method is to partition the vertex set of a graph randomly into two disjoint sets to separate a solution from the rest of the graph into connected components, and then select appropriate components to form a solution. We can use universal sets to derandomize algorithms obtained from this method. This new method is versatile and powerful as it can be used to solve a wide range of fixed-cardinality optimization problems for degree-bounded graphs, graphs of bounded degeneracy (a large family of graphs that contains degree-bounded graphs, planar graphs, graphs of bounded treewidth, and nontrivial minor-closed families of graphs), and even general graphs.

For a family F of graphs and a nonnegative integer k, F + ke and F − ke, respectively, denote the families of graphs that can be obtained from F graphs by adding and deleting at most k edges, and F + kv denotes the family of graphs that can be made into F graphs by deleting at most k vertices. This paper is mainly concerned with the parameterized complexity of the vertex colouring problem on F + ke, F − ke and F + kv for various families F of graphs. In particular, it is shown that the vertex colouring problem is ÿxed-parameter tractable (linear time for each ÿxed k) for split + ke graphs and split − ke graphs, solvable in polynomial time for each ÿxed k but W [1]-hard for split + kv graphs. Furthermore, the problem is solvable in linear time for bipartite + 1v graphs and bipartite + 2e graphs but, surprisingly, NP-complete for bipartite + 2v graphs and bipartite + 3e graphs.

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