Saturation number of lattice animals

On the saturation number of graphs

Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation numbe...

Statistics of lattice animals

The scaling behavior of randomly branched polymers in a good solvent is studied in two to nine dimensions, modeled by lattice animals on simple hypercubic lattices. For the simulations, we use a biased sequential sampling algorithm with resampling, similar to the pruned-enriched Rosenbluth method (PERM) used extensively for linear polymers. We obtain high statistics of animals with up to severa...

Induced saturation number

In this paper, we discuss the induced saturation number. It is a nice generalization of the saturation number that will allow us to consider induced subgraphs. We define the induced saturation number, indsat(n,H), to be the fewest number of gray edges in a trigraph T such that H does not appear in any realization of T , but if a black or white edge of T is flipped to gray then there exists a re...

Irredundance saturation number of a graph

Let G = (V,E) be a graph and let v ∈ V. Let IRS(v,G) denote the maximum cardinality of an irredundant set in G which contains v. Then IRS(G) = min {IRS(v,G) : v ∈ V } is called the irredundance saturation number of G. In this paper we initiate a study of this parameter.

Lattice animals and heaps of dimers