In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"-instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete. 1 Introduction This paper focuses on a number of graph decision problems which share the characteristic that all have a uniform upper bo...

This document is mainly based on "A Compendium of Parameterized Complexity Results", version 2.0 (May 22, 1996), by Michael T. Hallett and H. Todd Wareham, and on Downey and Fellows' book [53]. However, this document includes several new results that have been published in the last few years

Abstract. We study the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices with a bijection between its colors and the colors of the motif. This problem has applications in metabolic network analysis, an important area in bioinformatics. We give two positive results and three negative results that together draw sharp borderlines between tractable and intractable instances of the problem. 1

In this paper we review ways of representing incoherent 'knowledge' in a consistent way, where the use of modal logic and Kripke-style semantics is put central. Starting with a presentation of the basic modal framework, we discuss the basic modal systems K, KD (with an excursion to the representation of conflicting norms in deontic logic) and Chellas' minimal modal logic D. Next we look at the epistemic logics KD45, S4 and S5, including the logical omniscience problem and several non-standard modal logics to overcome this problem. After this we turn to the issue of reasoning by default, where a conflict of defaults (or default beliefs) may arise. We give an epistemic treatment of default reasoning, and treat the way conflicts of defaults can be solved viewed from the more general perspective of resolving conflicts in meta- level reasoning. Furthermore, special attention is paid to specificity in default reasoning as a principle to solve these conflicts, for which we develop an extension of Halpern & Moses' theory of honest formulas. Finally, we discuss several numerical modal logics in their capacity of ways of representation of incoherent information.

The problem of Intervalizing Colored Graphs (ICG) has received a lot of attention due to their use as a model for DNA physical mapping with ambiguous data. If k is the number of colors, the problem is known to be NP-Complete for general graphs for k 4 and has polynomial time algorithms for k = 2 and k = 3. In this paper we show that the ICG problem is NP-complete when the graph is a caterpillar tree, colored with k 4 colors, strengthen the cases for which the problem remains difficult. 1 Introduction Interval graphs have become very fashionable because they model the overlaps of DNA clones (see Section 5.2 in [SM97]). Recall that in an interval graph one can assign to each vertex in the graph an interval of the real line, in such a way that two vertices are adjacent if and only if their intervals have nonempty intersection [Gol80]. Given a graph with colors assigned to its vertices the problem of Intervalizing Colored Graphs (ICG) consists in deciding if there is a properly colored ...

The problem to determine whether a given k-colored graph is a subgraph of a properly colored interval graph has an application in DNA physical mapping. In this paper, we study the problem for the case that the number of colors k is fixed. For k = 2, we give a simple linear time algorithm, for k = 3, we give an O(n^2) algorithm for biconnected graphs with n vertices, and for k = 4, we show that the problem is NP-complete.

This report discusses some trends and achievements in computational geometry during the past five years, with emphasis on problems related to computer graphics. Furthermore, a direction of research in computational geometry is discussed, which could help in bringing the fields of computational geometry and computer graphics closer together.

In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded treewidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"-instances. For all of these problems with the exceptions of feasible register assignment and module allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 Z + . We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete.

In the Intervalizing Colored Graphs problem, one must decide for a given graph G = (V, E) with a proper vertex coloring of G whether G is the subgraph of a properly colored interval graph. For the case that the number of colors k is xed, we give an exact algorithm that uses O ∗ (2n/log1−ɛ(n)) time for all ɛ> 0. We also give an O ∗ (2n) algorithm for the case that the number of colors k is not xed. 1