Tree-decompositions with bags of small diameter

Tree-decompositions with bags of small diameter

This paper deals with the length of a Robertson–Seymour’s tree-decomposition. The tree-length of a graph is the largest distance between two vertices of a bag of a tree-decomposition, minimized over all tree-decompositions of the graph. The study of this invariant may be interesting in its own right because the class of bounded tree-length graphs includes (but is not reduced to) bounded chordal...

Tree Decompositions with Small Cost

The f -cost of a tree decomposition (fXi j i 2 Ig; T = (I; F )) for a function f : N ! R+ is de ned as P i2I f(jXij). This measure associates with the running time or memory use of some algorithms that use the tree decomposition. In this paper we investigate the problem to nd tree decompositions of minimum f -cost. A function f : N! R+ is fast, if for every i 2 N: f(i+1) 2 f(i). We show that fo...

Decompositions into Subgraphs of Small Diameter

We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P (n, 2, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ . . . ∪ Ek such that |E0| ≤ 2n and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi’s regularity lemma, Polcyn and Ruciński showed that P (n, 2, 4) is...

Tree-decompositions of small pathwidth

Tree spanners of small diameter

A graph that contains a spanning tree of diameter at most t clearly admits a tree t-spanner, since a tree t-spanner of a graph G is a sub tree of G such that the distance between pairs of vertices in the tree is at most t times their distance in G. In this paper, graphs that admit a tree t-spanner of diameter at most t + 1 are studied. For t equal to 1 or 2 the problem has been solved. For t = ...