TY - GEN A1 - Wolle, Thomas A1 - Koster, Arie M.C.A. A1 - Bodlaender, Hans L. T1 - A Note on Contraction Degeneracy N2 - The parameter contraction degeneracy -- the maximum minimum degree over all minors of a graph -- is a treewidth lower bound and was first defined in (Bodlaender, Koster, Wolle, 2004). In experiments it was shown that this lower bound improves upon other treewidth lower bounds. In this note, we examine some relationships between the contraction degeneracy and connected components of a graph, block s of a graph and the genus of a graph. We also look at chordal graphs, and we study an upper bound on the contraction degeneracy and another lower bound for treewidth. A data structure that can be used for algorithms computing the degeneracy and similar parameters, is also described. T3 - ZIB-Report - 04-43 KW - contraction degeneracy KW - genus of a graph KW - treewidth lower bounds Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8180 ER - TY - GEN A1 - Koster, Arie M.C.A. A1 - Wolle, Thomas A1 - Bodlaender, Hans L. T1 - Degree-Based Treewidth Lower Bounds N2 - Every lower bound for treewidth can be extended by taking the maximum of the lower bound over all subgraphs or minors. This extension is shown to be a very vital idea for improving treewidth lower bounds. In this paper, we investigate a total of nine graph parameters, providing lower bounds for treewidth. The parameters have in common that they all are the vertex-degree of some vertex in a subgra ph or minor of the input graph. We show relations between these graph parameters and study their computational complexity. To allow a practical comparison of the bounds, we developed heuristic algorithms for those parameters that are NP-hard to compute. Computational experiments show that combining the treewidth lower bounds with minors can considerably improve the lower bounds. T3 - ZIB-Report - 04-44 KW - contraction degeneracy KW - Ramachandramurthi parameter KW - treewidth lower bounds Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8193 ER - TY - GEN A1 - Bodlaender, Hans L. A1 - Koster, Arie M.C.A. T1 - On the Maximum Cardinality Search Lower Bound for Treewidth N2 - The Maximum Cardinality Search algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbors becomes visited. An MCS-ordering of a graph is an ordering of the vertices that can be generated by the Maximum Cardinality Search algorithm. The visited degree of a vertex $v$ in an MCS-ordering is the number of neighbors of $v$ that are before $v$ in the ordering. The visited degree of an MCS-ordering $\psi$ of $G$ is the maximum visited degree over all vertices $v$ in $\psi$. The maximum visited degree over all MCS-orderings of graph $G$ is called its {\em maximum visited degree}. Lucena (2003) showed that the treewidth of a graph $G$ is at least its maximum visited degree. We show that the maximum visited degree is of size $O(\log n)$ for planar graphs, and give examples of planar graphs $G$ with maximum visited degree $k$ with $O(k!)$ vertices, for all $k\in \Bbb{N}$. Given a graph $G$, it is NP-complete to determine if its maximum visited degree is at least $k$, for any fixed $k\geq 7$. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete. We also propose and experimentally analyses some heuristics for the problem. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications. T3 - ZIB-Report - 04-45 KW - maximum cardinality search KW - treewidth KW - lower bounds KW - planar graphs KW - graph algorithms Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8201 ER -