We study the parameterized complexity of three NP-hard graph completion problems. The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the big-O notation are small and do not depend on k. In particular, this implies that the problem is fixed-parameter tractable (FPT). The PROPER

In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G² of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a dually chordal graph, then there exists a spanning tree T of G that is an additive 3-spanner as well as a multiplicative 4-spanner of G. In all cases the tree T can be computed in linear time

A graph G is a circular-arc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a linear-time algorithm for recognizing this class of graphs. When G is a member of the class, the algorithm gives a certificate in the form of a set of arcs that realize it.

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(|V|) time.

Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified.

An interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each pair of intersecting intervals. A probe interval graph is obtained from an interval graph by designating a subset P of vertices as probes, and removing the edges between pairs of vertices in the remaining set N of non-probes. We examine the problem of finding and representing possible layouts of the intervals, given a probe interval graph. We obtain an O(n + m log n) bound, where n is the number of vertices and m is the number of edges. The problem is motivated by an application to molecular biology.

Partition refinement techniques lead to simple and efficient algorithms for various applications: automaton minimization, string sorting . . . and also for algorithms on graphs. A generic algorithm that can be used for all these applications is presented and briefly discussed. Such an approach is interesting in an algorithmic tool kit perspective. New instances of the generic algorithm are presented: O(n + m log n) very simple and practical algorithms for cographs recognition and for modular decomposition are designed. Although these algorithms are not linear, they are very easy to implement (as an instanciation of a generic procedure) and based on new ideas.

An extremely simple, linear time algorithm is given for constructing a domatic partition in totally balanced hypergraphs. This simplifies and generalizes previous algorithms for interval and strongly chordal graphs. On the other hand, the domatic number problem is shown to be NP-complete for several families of perfect graphs, including chordal and bipartite graphs.

Feodor F.Dragan Dept. ofCompu6; Science, Kent State University, Kent, Ohio 44242, USA dragan@cs.k Abstract. In this paper we show that a very simple and e#cient approach can beu=6 to solve the all pairs almostshost problem on the class of weakly chordal graphs and its di#erentsu classes. Moreover, this approach works well also on graphs with small size of largest indust cycle and gives au=6; way to solve the all pairs almostshost and all pairsshs problems on di#erent graph classes inclus ing chordal, strongly chordal, chordal bipartite, and distance-hereditary graphs. 1

Let be a family of cliques of a graph G =(V,E). Suppose that each clique C of is associated with an integer r(C), where r(C) 0. A vertex vr-dominates a clique C of G if d(v, x) r(C) for all x C, where d(v, x) is the standard graph distance. A subset D V is a clique r-dominating set of G if for every clique C is a vertex u D which r-dominates C. A clique r-packing set is a subset P that there are no two distinct cliques C # ,C ## Pr-dominated by a common vertex of G. The clique r-domination problem is to find a clique r-dominating set with minimum size and the clique r-packing problem is to find a clique r-packing set with maximum size. The formulated problems include many domination and clique-transversal-related problems as special cases. In this paper an efficient algorithm is proposed for solving these problems on dually chordal graphs which are a natural generalization of strongly chordal graphs. The efficient algorithm is mainly based on the tree structure and special vertex elimination orderings of dually chordal graphs. In some important particular cases where the algorithm works in linear time the obtained results generalize and improve known results on strongly chordal graphs.