An undirected graph G = (V, E) is a probe split graph if its vertex set can be partitioned into two sets, N (nonprobes) and P (probes) where N is independent and there exists E′ ⊆ N ×N such that G′ = (V, E ∪ E′) is a split graph. Recently Chang et al. gave an O(V (V +E)) time recognition algorithm for probe split graphs. In this article we give O(V 2 +V E) time recognition algorithms and charac...

Recently Postnikov gave a combinatorial description of the cells in a totallynonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. We prove his conjecture that a positroid is exactly an intersection of permuted Schubert matroids. This leads to a combinatorial description of positroids that is easily computable. The main proof is purely combinatorial,...

The paper shows that there is no finite set of forbidden minors which characterizes classes of independence models that are representable by Gaussian, discrete and binary distributions, respectively. In addition, a way to prove the nonexistence of a finite characterization for any other class of independence models is suggested. MSC 2000: 60A99, 68T30, 94A15

We show that the norm graph constructed in [13] with n vertices about 1 2n 2−1/t edges, which contains no copy of Kt,(t−1)!+1, does not contain a copy of Kt+1,(t−1)!−1.

In this paper we investigate the closure ~-* under substitution-composition of a family of graphs ~,, defined by a set Lr of forbidden configurations. We first prove that ~-* can be defined by a set L~* of forbidden subgraphs. Next, using a counterexample we show that ~* is not necessarily a finite set, even when .~ is finite. We then give a sufficient condition for ~* to be finite and a simple...

We characterise t-perfect plane triangulations by forbidden induced subgraphs. As a consequence, we obtain that a plane triangulation is h-perfect if and only if it is perfect.