[1, 2]-sets and [1, 2]-total Sets in Trees with Algorithms

A set $S \subseteq V$ of the graph $G = (V, E)$ is called a $[1, 2]$-set of $G$ if any vertex which is not in $S$ has at least one but no more than two neighbors in $S$. A set $S \subseteq V$ is called a $[1, 2]$-total set of $G$ if any vertex of $G$, no matter in $S$ or not, is adjacent to at least one but not more than two vertices in $S$. In this paper we introduce a linear algorithm for finding the cardinality of the smallest $[1, 2]$-sets and $[1, 2]$-total sets of a tree and extend it to a more generalized version for $[i, j]$-sets, a generalization of $[1, 2]$-sets. This answers one of the open problems proposed in [5]. Then since not all trees have $[1, 2]$-total sets, we devise a recursive method for generating all the trees that do have such sets. This method also constructs every $[1, 2]$-total set of each tree that it generates.