5 research outputs found
Independent [1,2]-number versus independent domination number
A [1, 2]-set S in a graph G is a vertex subset such that every vertex
not in S has at least one and at most two neighbors in it. If the additional
requirement that the set be independent is added, the existence of such
sets is not guaranteed in every graph. In this paper we provide local
conditions, depending on the degree of vertices, for the existence of
independent [1, 2]-sets in caterpillars. We also study the relationship
between independent [1, 2]-sets and independent dominating sets in this
graph class, that allows us to obtain an upper bound for the associated
parameter, the independent [1, 2]-number, in terms of the independent
domination number.Peer ReviewedPostprint (published version
On Independent [1, 2]-Sets in Trees
An [1, k]-set S in a graph G is a dominating set such that every vertex not in S has at most k neighbors in it. If the additional requirement that the set must be independent is added, the existence of such sets is not guaranteed in every graph. In this paper we solve some problems previously posed by other authors about independent [1, 2]-sets. We provide a necessary condition for a graph to have an independent [1, 2]-set, in terms of spanning trees, and we prove that this condition is also sufficient for cactus graphs. We follow the concept of excellent tree and characterize the family of trees such that any vertex belongs to some independent [1, 2]-set. Finally, we describe a linear algorithm to decide whether a tree has an independent [1, 2]-set. This algorithm can be easily modified to obtain the cardinality of a smallest independent [1, 2]-set of a tree
On Independent [1, 2]-Sets in Trees
An [1, k]-set S in a graph G is a dominating set such that every vertex not in S has at most k neighbors in it. If the additional requirement that the set must be independent is added, the existence of such sets is not guaranteed in every graph. In this paper we solve some problems previously posed by other authors about independent [1, 2]-sets. We provide a necessary condition for a graph to have an independent [1, 2]-set, in terms of spanning trees, and we prove that this condition is also sufficient for cactus graphs. We follow the concept of excellent tree and characterize the family of trees such that any vertex belongs to some independent [1, 2]-set. Finally, we describe a linear algorithm to decide whether a tree has an independent [1, 2]-set. This algorithm can be easily modified to obtain the cardinality of a smallest independent [1, 2]-set of a tree
Independent [1,2]-number versus independent domination number
A [1, 2]-set S in a graph G is a vertex subset such that every vertex
not in S has at least one and at most two neighbors in it. If the additional
requirement that the set be independent is added, the existence of such
sets is not guaranteed in every graph. In this paper we provide local
conditions, depending on the degree of vertices, for the existence of
independent [1, 2]-sets in caterpillars. We also study the relationship
between independent [1, 2]-sets and independent dominating sets in this
graph class, that allows us to obtain an upper bound for the associated
parameter, the independent [1, 2]-number, in terms of the independent
domination number.Peer Reviewe