Young domination on Hamming rectangles

Abstract: In the neighborhood growth dynamics on a Hamming rectangle $[0,m-1]\times[0,n-1]\subseteq \mathbb{Z}_+^2$, the decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. After the initially occupied set is chosen, the synchronous rule is iterated. The Young domination number with a fixed latency $L$ is the smallest cardinality of an initial set that covers the rectangle by $L$ steps, for $L=0,1,\ldots$ We compute this number for some special cases, including $k$-domination for any $k$ when $m=n$, thereby proving a conjecture from 2009 due to Burchett, Lachniet, and Lane, and devise approximation algorithms in the general case. These results have implications in extremal graph theory, via an equivalence between the case $L = 1$ and bipartite Tur\'an numbers for families of double stars. Our approach is based on a variety of techniques including duality between Young diagrams, algebraic formulations, explicit constructions, and dynamic programming.