We present a pumping lemma for the class of collapsible pushdown graphs of level 2. This pumping lemma even applies to the ε-contractions of level 2 collapsible pushdown graphs. Our pumping lemma also improves the bounds of Hayashi’s pumping lemma for indexed languages. 1998 ACM Subject Classification F.1.1 Models of Computation

Jha, Pradhan, and Banerjee devised a linear algorithm to compute the secure domination number of cograph. Here it is shown that their Lemma 2, which crucial for computational complexity algorithm, incomplete. An accordingly modified lemma proved demonstrated remains linear.

Part II of this study uses the path-following theory of labelled V-complexes developed in Part I to provide constructive algorithmic proofs of a variety of combinatorial lenmas in topology. \Je demonstrate two new dual lemmas on the n-dimensional cube, and use a Generc.lized S ^erner Lemma to prove a gener lization of the Knaster-Kuratowski-y.azurkiewicz Covering Lemma on the simplex. We also s...

Let D(n)(D(n, k)) denote the maximum possible d such that there exists a d-regular hypergraph (d-regular k-uniform hypergraph, respectively) on n vertices containing no proper regular spanning subhypergraph. The problem of estimating D(n) arises in Game Theory and Huckemann and Jurkat were the first to prove that it is finite. Here we give two new simple proofs that D(n), D(n, k) are finite, an...

expressed in the lemma/palea would provide molecular clues to the functions of these organs. The lemma and palea (lemma/palea) of cereals are photosynthetic Loss-of-function studies and cloning of floral homeoorgans that supply the developing kernel with carbon and nitrogen. tic genes suggest that the lemma/palea of grasses are Because of their rigid structure, the lemma/palea can also protect ...

The celebrated S-lemma establishes a powerful equivalent condition for the nonnegativity of a quadratic function over a single quadratic inequality. However, this lemma fails without the technical condition, known as the Slater condition. In this paper, we first show that the Slater condition is indeed necessary for the S-lemma and then establishes a regularized form of the S-lemma in the absen...

Szemerédi’s regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemerédi’s theorem on arithmetic progressions [19], [18]. In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to ...

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovász Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.