We introduce a family of domination-type problems in Cartesian products of two graphs. The framework captures several well-studied topics, including variants of bootstrap percolation, line growth, distance domination, and target set selection. We focus on Cartesian products of two complete graphs and formulate the notion of Young domination number in terms of a growth rule determined by a Young diagram; this number is the smallest cardinality of an initial set that covers the entire vertex set in a prescribed number $L$ of iterations of the rule. We compute the Young domination number with $L=1$ for several natural cases, including $k$-domination for Cartesian products of two complete graphs of the same order, thereby proving a conjecture from 2009 due to Burchett, Lane, and Lachniet. We show that the case of $L=1$ of Young domination is equivalent to computing bipartite Turán numbers for families of double stars, yielding implications of our results in extremal graph theory. For arbitrary fixed $L$, we devise constant-factor approximation algorithms for the problem. Our approach is based on a variety of techniques, including duality between Young diagrams, algebraic formulations, explicit constructions, and dynamic programming.

翻译：本文引入了一类在图的笛卡尔积上的支配型问题。该框架涵盖多个已有深入研究的方向，包括自举渗流变体、线增长、距离支配以及目标集选择等。我们聚焦于两个完全图的笛卡尔积，并借助由杨图决定的增长规则，形式化地定义了Young支配数的概念；该数是指一个初始集合的最小基数，使得在规则迭代$L$次后能覆盖整个顶点集。我们计算了$L=1$时若干自然情形下的Young支配数，包括同阶两个完全图笛卡尔积的$k$-支配，从而证明了Burchett、Lane和Lachniet于2009年提出的一个猜想。我们证明了Young支配在$L=1$的情形等价于计算双星图族的二分图Turán数，这使我们的结果在极值图论中具有推论意义。对于任意固定的$L$，我们设计了该问题的常数因子近似算法。我们的方法基于多种技术，包括杨图的对偶性、代数形式化、显式构造以及动态规划。