173 research outputs found
Total irredundance in graphs
AbstractA set S of vertices in a graph G is called a total irredundant set if, for each vertex v in G,v or one of its neighbors has no neighbor in S−{v}. We investigate the minimum and maximum cardinalities of maximal total irredundant sets
k-Tuple_Total_Domination_in_Inflated_Graphs
The inflated graph of a graph with vertices is obtained
from by replacing every vertex of degree of by a clique, which is
isomorph to the complete graph , and each edge of is
replaced by an edge in such a way that , , and
two different edges of are replaced by non-adjacent edges of . For
integer , the -tuple total domination number of is the minimum cardinality of a -tuple total dominating set
of , which is a set of vertices in such that every vertex of is
adjacent to at least vertices in it. For existing this number, must the
minimum degree of is at least . Here, we study the -tuple total
domination number in inflated graphs when . First we prove that
, and then we
characterize graphs that the -tuple total domination number number of
is or . Then we find bounds for this number in the
inflated graph , when has a cut-edge or cut-vertex , in terms
on the -tuple total domination number of the inflated graphs of the
components of or -components of , respectively. Finally, we
calculate this number in the inflated graphs that have obtained by some of the
known graphs
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
A Note on Isolate Domination
A set of vertices of a graph such that has an isolated vertex is called an \emph{isolate set} of . The minimum and maximum cardinality of a maximal isolate set are called the \emph{isolate number} and the \emph{upper isolate number} respectively. An isolate set that is also a dominating set (an irredundant set) is an \emph{isolate dominating set} \ (\emph{an isolate irredundant set}). The \emph{isolate domination number} and the \emph{upper isolate domination number} are respectively the minimum and maximum cardinality of a minimal isolate dominating set while the \emph{isolate irredundance number} and the \emph{upper isolate irredundance number} are the minimum and maximum cardinality of a maximal isolate irredundant set of . The notion of isolate domination was introduced in \cite{sb} and the remaining were introduced in \cite{isrn}. This paper further extends a study of these parameters
FROM IRREDUNDANCE TO ANNIHILATION: A BRIEF OVERVIEW OF SOME DOMINATION PARAMETERS OF GRAPHS
Durante los últimos treinta años, el concepto de dominación en grafos ha levantado un interés impresionante. Una bibliografÃa reciente sobre el tópico contiene más de 1200 referencias y el número de definiciones nuevas está creciendo continuamente. En vez de intentar dar un catálogo de todas ellas, examinamos las nociones más clásicas e importantes (tales como dominación independiente, dominación irredundante, k-cubrimientos, conjuntos k-dominantes, conjuntos Vecindad Perfecta, ...) y algunos de los resultados más significativos.  PALABRAS CLAVES: TeorÃa de grafos, Dominación.  ABSTRACT During the last thirty years, the concept of domination in graphs has generated an impressive interest. A recent bibliography on the subject contains more than 1200 references and the number of new definitions is continually increasing. Rather than trying to give a catalogue of all of them, we survey the most classical and important notions (as independent domination, irredundant domination, k-coverings, k-dominating sets, Perfect Neighborhood sets, ...) and some of the most significant results.  KEY WORDS: Graph theory, Domination
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