Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the forbidden number. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.

In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs o...

Forbidden minors and subdivisions for toroidal graphs are numerous. In contrast, the toroidal graphs with no K3,3’s have a nice explicit structure and short lists of obstructions. For these graphs, we provide the complete lists of four forbidden minors and eleven forbidden subdivisions.

A graph is 1-extendible if every edge has a 1-factor containing it. A 1-extendible non-bipartite graph G is said to be near bipartite if there exist edges e1 and e2 such that G − {e1, e2} is 1-extendible and bipartite. We characterise the Pfaffian near bipartite graphs in terms of forbidden subgraphs. The theorem extends an earlier characterisation of Pfaffian bipartite graphs.

We consider the problem of indexing a collection of documents (a.k.a. strings) of total length n such that the following kind of queries are supported: given two patterns P and P−, list all nmatch documents containing P but not P−. This is a natural extension of the classic problem of document listing as considered by Muthukrishnan [SODA’02], where only the positive pattern P is given. Our main...

The twin of this paper, “Forbidden Magnification? I.” [1], presents systematic SOM simulations with the explicit magnification control scheme of Bauer, Der, and Herrmann [2] on data for which the theory does not guarantee success, namely data that are n-D, n > 2 and/or data whose components in the different dimensions are not statistically independent. For the unsupported n = 2 cases that we in...

The world of crystallography was forced to reassess its rules about thirty years ago with the introduction of the concept of quasicrystals, solids with rotational symmetries forbidden to crystals, by Levine and Steinhardt (1984) and the discovery of the first examples in the laboratory by Shechtman et al. (1984). Since then, >100 different types of quasicrystals have been synthesized in the lab...

Toriyeh means concealing real intention of speech using its parallel and common words so that the listener constructs from speaker's speech a meaning what he/she meant. The purpose of this research is studying jurisprudential dimensions of toriyeh in order to clarify its distinction from lying and related jurisprudential commandments by explanation of the most important discussions about toriye...