On the residual solvability of generalized free products of solvable groups

In this paper, we study the residual solvability of the generalized free product of solvable groups.Comment: Discrete Mathematics & Theoretical Computer Science, Vol 13, 201

Encoding Arguments

Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is effective, but the underlying probabilistic machinery can be daunting. "Encoding arguments" provide an alternative presentation in which probabilistic reasoning is encapsulated in a "uniform encoding lemma". This lemma provides an upper bound on the probability of an event using the fact that a uniformly random choice from a set of size $n$ cannot be encoded with fewer than $\log_2 n$ bits on average. With the lemma, the argument reduces to devising an encoding where bad objects have short codewords. In this expository article, we describe the basic method and provide a simple tutorial on how to use it. After that, we survey many applications to classic problems from discrete mathematics and computer science. We also give a generalization for the case of non-uniform distributions, as well as a rigorous justification for the use of non-integer codeword lengths in encoding arguments. These latter two results allow encoding arguments to be applied more widely and to produce tighter results.Comment: 50 pages, 7 figure

On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007

2×22\times 2 monotone grid classes are finitely based

In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row.Comment: 10 pages, 5 figures. To appear in Discrete Mathematics and Theoretical Computer Science, special issue for Permutation Patterns 201

Heredity for generalized power domination

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma\_{p,k}(G-e)$, $\gamma\_{p,k}(G/e)$ and for $\gamma\_{p,k}(G-v)$ in terms of $\gamma\_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma\_{p,k}(G-e) = \gamma\_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.Comment: Discrete Mathematics and Theoretical Computer Science, 201