Upper $k$-Star-Forming Sets, $k$-Independence, and Upper Domination

Abstract

For a positive integer $k$, let $\beta_k(G)$ be the maximum cardinality of a vertex set inducing maximum degree less than $k$, and let $SF_k(G)$ be the maximum cardinality of a minimal $k$-star-forming set. The known inequality $\beta_k(G)\le SF_k(G)$ suggests asking when equality holds. We place this question in the framework of upper domination: at $k=1$, $\beta_1(G)=\alpha(G)$ and $SF_1(G)=\Gamma(G)$, so the classical equality $\Gamma=\alpha$ on bipartite graphs is exactly the first member of the proposed hierarchy. We prove the equality for complete bipartite graphs for every $k$, obtaining \[ \beta_k(K_{a,b})=SF_k(K_{a,b})=\max\{a,b,2k-2\}\qquad(a,b\ge k), \] and record the elementary low-degree case $\Delta(G)<k$. For $k=2$ we derive certificate restrictions for minimal $2$-star-forming sets in bipartite graphs. For chain graphs we go further: we prove $\beta_2(G)=SF_2(G)$ and obtain an exact formula for their common value. The proof uses the nested-neighborhood structure together with a classification of witnesses to the indispensability of a high internal-degree vertex. We retain the equality problem for chain graphs as a conjecture only for $k\ge3$, and formulate the broader bipartite conjecture. We also determine the upper domination number of every rectangular grid and combine it with the known exact dissociation number to compare $\Gamma$, $\beta_2$, and $SF_2$. In particular, $\Gamma=\beta_2$ on every even-by-even rectangular grid, while $\beta_2\le SF_2$ always; this motivates a grid equality conjecture whose even-by-even case would yield a three-parameter identity.