Self-stabilizing algorithm for minimal (α,β)-dominating set

A self‐stabilizing distributed algorithm for the bounded lattice domination problems under the distance‐2 model

“…If we adopt “$$$ >1 $$$” instead, $$$ C $$$ would not be closed. It means that, in the function $$$ fmin(i) $$$ in Algorithm 1 presented later, there exists no minimum value so that the domination is maintained, and there exists no minimal domination. $$$ \alpha $$$‐domination (MADS)A set $$$ S\subseteq V $$$ is an $$$ \alpha $$$ ‐dominating set 26 iff $$$ \mid S\cap N(u)\mid /\mid N(u)\mid \ge \alpha $$$ for each $$$ u\in V-S $$$. Here, $$$ \alpha $$$ is a real number such that $$$ 0<\alpha \le 1 $$$.…”

“…A trivial solution is $$$ f(u)=1 $$$ for each $$$ u\in V $$$. $$$ U\equiv \left\{0,1\right\} $$$. $$$ C\left({g}_0,{g}_1,\dots, {g}_d\right)\equiv \left({g}_0=0\right)\wedge \left({\sum}_{\ell =1}^d{g}_{\ell }/d\ge \alpha \right)\vee \left({g}_0=1\right) $$$. $$$ \left(\alpha, \beta \right) $$$‐domination (MABDS) when $$$ \alpha \ge \beta $$$A set $$$ S\subseteq V $$$ is an $$$ \left(\alpha, \beta \right) $$$ ‐dominating set 26 iff $$$ \mid S\cap N(u)\mid /\mid N(u)\mid \ge \alpha $$$ for each $$$ u\in V-S $$$, and …”

“…A set $$$ S\subseteq V $$$ is an $$$ \alpha $$$ ‐dominating set 26 iff $$$ \mid S\cap N(u)\mid /\mid N(u)\mid \ge \alpha $$$ for each $$$ u\in V-S $$$. Here, $$$ \alpha $$$ is a real number such that $$$ 0<\alpha \le 1 $$$.…”

“…A set $$$ S\subseteq V $$$ is an $$$ \left(\alpha, \beta \right) $$$ ‐dominating set 26 iff $$$ \mid S\cap N(u)\mid /\mid N(u)\mid \ge \alpha $$$ for each $$$ u\in V-S $$$, and $$$ \mid S\cap N(u)\mid /\mid N(u)\mid \ge \beta $$$ for each $$$ u\in S $$$. Here, $$$ \alpha $$$ (resp., $$$ \beta $$$) is a real number such that $$$ 0<\alpha \le 1 $$$ (resp., $$$ 0\le \beta \le 1 $$$).…”

“…Here, a total $$$ k $$$‐dominating set $$$ D $$$ is a total dominating set, and each vertex has at least $$$ k $$$ neighbors in $$$ D $$$. In Reference 26, Saadi et al proposed self‐stabilizing distributed algorithms for the $$$ \alpha $$$‐dominating set problem and the $$$ \left(\alpha, \beta \right) $$$‐dominating set problem. These problems require the minimum ratio of the number of dominators in neighbors, whereas the $$$ k $$$‐dominating set problem requires the minimum number of dominators in neighbors.…”