Bernhard Fuchs

Covering Graphs by Colored Stable Sets

Abstract Let G = ( V , R ∪ B ) be a multigraph with red and blue edges. G is an R / B -split graph if V is the union of a red and a blue stable set. R / B -split graphs yield a common generalization of split graphs and Konig graphs. It is shown, for example, that R / B -split graphs can be recognized in polynomial time. On the other hand, finding a maximal R / B -subgraph is NP -hard already for the class of comparability graphs of series-parallel orders. Moreover, there can be no approximation ratio better than 31/32 unless P = NP .

The Number of Tree Stars Is O * (1.357 k )

Abstract Every rectilinear Steiner tree problem admits an optimal tree T* which is composed of tree stars. Moreover, the currently fastest algorithms for the rectilinear Steiner tree problem proceed by composing an optimum tree T* from tree star components in the cheapest way. The efficiency of such algorithms depends heavily on the number of tree stars (candidate components). Fößmeier and Kaufmann (Algorithmica 26, 68–99, 2000) showed that any problem instance with k terminals has a number of tree stars in between 1.32k and 1.38k (modulo polynomial factors) in the worst case. We determine the exact bound O*(ρk) where ρ≈1.357 and mention some consequences of this result.

The number of tree stars is O*(1.357k)

Every rectilinear Steiner tree problem admits an optimal tree T* which is composed of tree stars. Moreover, the currently fastest algorithms for the rectilinear Steiner tree problem proceed by composing an optimum tree T* from tree star components in the cheapest way. The efficiency of such algorithms depends heavily on the number of tree stars (candidate components). Fosmeier and Kaufmann [U. Fosmeier, M. Kaufmann, On exact solutions for the rectilinear Steiner tree problem Part 1: Theoretical results, Algorithmica 26 (2000) 68–99] showed that any problem instance with k terminals has a number of tree stars in between 1.32k and 1.38k (modulo polynomial factors) in the worst case. We determine the exact bound of O∗(αk) where α≈1.357 and mention some consequences of this result.