Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal labeling, all solutions and all realizations problems are considered for temporal (interval) data. Several versions are investigated by restricting the possible interval relationships yielding different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NP-complete. On the positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NP-complete. This problem is als...

Given a family of sets S, where the sets in S admit k ‘degrees of freedom’, we prove that not all (k + 1)-dimensional posets are containment posets of sets in S. Our results depend on the following enumerative result of independent interest: Let P (n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P (n, k) ∼ nk log n where k is fixed and n is large. KEY WORDS: partially ordered set, containment order, degrees of freedom, partial order dimension AMS SUBJECT CLASSIFICATION: 06A10 (primary), 14N10 (secondary) 1.

A graph property (i.e., a set of graphs) is induced-hereditary or additive if it is closed under taking induced-subgraphs or disjoint unions. If P and Q are properties, the product P ◦ Q consists of all graphs G for which there is a partition of the vertex set of G into (possibly empty) subsets A and B with G[A] ∈ P and G[B] ∈ Q. A property is ∗ The results presented here are part of the first author’s Ph.D. thesis, written under the supervision of the second author. Jim Geelen suggested one of the main results of this paper. † The first author’s doctoral studies in Canada were fully funded by the Canadian government through a Canadian Commonwealth Scholarship. ‡ Research supported by NSERC.

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A graph G is a subtree overlap graph if there exists a tree T and a set of subtrees fT i g so that there exists a one-to-one mapping between vertices and subtrees and two subtrees overlap if and only if their respective vertices are adjacent. The class of subtree overlap graphs is proven to contain the classes of circle, spider or circle polygon, and chordal graphs. An upper bound on the size of the subtree overlap model is proven to be 3m. As well a general algorithm to find the maximum independent set for any class of overlap graph is given, provided testing for containment and intersection in the overlap graph can be done in polynomial time, and the maximumweight independent set problem is solved for the related class of intersection graph. The complexities of the Hamiltonian Cycle, several domination problems, isomorphism and colouring are shown to be as hard for subtree overlap graphs as they are for graphs in general. Acknowledgements Many people have helped me with this thesi...

Containment and overlap representations of digraphs are studied, with the following results. The interval containment digraphs are the digraphs of Ferrers dimension 2, and the circular-arc containment digraphs are the complements of circular-arc intersection digraphs. A poset is an interval containment poset if and only if its comparability digraph is an interval containment digraph, and a graph is an interval graph if and only if the corresponding symmetric digraph with loops is an interval digraph. In an appropriate model of overlap representation using intervals, the unit right-overlap interval digraphs are precisely the unit interval digraphs, and the adjacency matrices of right-overlap interval digraphs have a simple structural characterization bounding their Ferrers dimension by 3. Keywords: digraph, intersection representation, containment, overlap, unit interval, Ferrers dimension Running head: REPRESENTATIONS OF DIGRAPHS y Research supported in part by NSA/MSP Grant MDA904-9...