Eberhard Triesch

Characterisation of Archimedian t -norms and a law of large numbers

Abstract We study a law of large numbers for mutually T -related fuzzy numbers where T is an Archimedian t-norm and extend earlier results of Fuller in this area. In particular, we show that the class of Archimedian t-norms can be characterised by the validity of a very general law of large numbers for sequences of L-R fuzzy numbers.

Superdominance order and distance of trees with bounded maximum degree

Using a weaker variant of the well-known dominance order on partitions, we determine the trees of fixed order and bounded maximum degree that have minimum distance.

Realizability and uniqueness in graphs

Abstract Consider a finite graph G ( V , E ). Let us associate to G a finite list P ( G ) of invariants. To any P the following two natural problems arise: (R) Realizability. Given P , when is P = P ( G ) for some graph G ?, (U) Uniqueness. Suppose P ( G )= P ( H ) for graphs G and H . When does this imply G ≅ H ? The best studied questions in this context are the degree realization problem for (R) and the reconstruction conjecture for (U). We discuss the problems (R) and (U) for the degree sequence and the size sequence of induced subgraphs for undirected and directed graphs, concentrating on the complexity of the corresponding decision problems and their connection to a natural search problem on graphs.

A group testing problem for hypergraphs of bounded rank

Abstract Suppose that a hypergraph H = (V, E) of rank r is given as well as a probability distribution p(e) (e ϵ E) on the edges. We show that in the usual group testing model the unknown edge can be found by less than — log p(e) + r tests. For the case of the uniform distribution, the result proves a conjecture of Du and Hwang.

Improved Results for Competitive Group Testing

We consider algorithms for group testing problems when nothing is known in advance about the number of defectives. Du and Hwang suggested measuring the quality of such algorithms by its so-called (first) competitive ratio (see the Introduction). Later, Du and Park suggested a second kind of competitive ratio. For each kind of competitiveness, we improve the best-known bounds: in the first case, from 1.65 to

On the convergence of product-sum series of L-R fuzzy numbers

1.5+\ep

Edge search in graphs and hypergraphs of bounded rank

, and in the second from 16 to 4.

Reconstructing a Graph from its Neighborhood Lists

We study series a1 + a2 + … + an + … of L-R fuzzy numbers where addition is defined via the sup-product-norm convulation and obtain some convergence theorems which generalize earlier results by Fuller (cf. [2]).

Irregular assignments and vertex distinguishing edge-colorings of graphs

Abstract Given a finite graph G =( V,E ), what is the worst-case complexity L ( G ) of finding an unknown edge e ∗ ϵ E if the following tests are admitted: For any W ⊂ V we may test whether e ∗ ⊂ W or not. We prove that L ( G )⩽log 2 | E |+3. This result is generalized to hypergraphs H =( V ; E ) of bounded rank: For any r , there exists some constant γ r such that L ( H )⩽log 2 | E |+γ r for any hypergraph with rank ⩽ r .

On the recognition complexity of some graph properties

Associate to a finite labeled graph G ( V, E ) its multiset of neighborhoods ( G ) = {N(υ): υ ∈ V }. We discuss the question of when a list is realizable by a graph, and to what extent G is determined by ( G ). The main results are: the decision problem is NP-complete; for bipartite graphs the decision problem is polynomially equivalent to Graph Isomorphism; forests G are determined up to isomorphism by ( G ); and if G is connected bipartite and ( H ) = ( G ), then H is completely described.