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 bibliografía 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: Teoría 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

On the bipartite independence number of a balanced bipartite graph

AbstractThe bipartite independence number αBIP of a bipartite graph G is the maximum order of a balanced independent set of G. Let δ be the minimum degree of the graph. When G itself is balanced, we establish some relations between αBIP and the size or the connectivity of G. We also prove that the condition αBIP⩽δ(resp.αBIP⩽δ−1) implies that G is hamiltonian (resp. Hamilton-biconnected), thus improving a result of Fraisse

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

On k-factor-critical graphs

A graph is said to be k-factor-critical if the removal of any set of k vertices results in a graph with a perfect matching. We study some properties of k-factor-critical graphs and show that many results on q-extendable graphs can be improved using this concept

Partial Domination and Irredundance Numbers in Graphs

A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted by $\gamma(G)$. A vertex set $D$ is a $k$-isolating set of $G$ if $G - N_{G}[D]$ contains no $k$-cliques. The minimum cardinality of a $k$-isolating set of $G$ is called the $k$-isolation number of $G$ and is denoted by $\iota_{k}(G)$. Clearly, $\gamma(G) = \iota_{1}(G)$. A vertex set $I$ is irredundant if, for every non-isolated vertex $v$ of $G[I]$, there exists a vertex $u$ in $V \setminus I$ such that $N_{G}(u) \cap I = \{v\}$. An irredundant set $I$ is maximal if the set $I \cup \{u\}$ is no longer irredundant for any $u \in V(G) \setminus I$. The minimum cardinality of a maximal irredundant set is called the irredundance number of $G$ and is denoted by $ir(G)$. Allan and Laskar \cite{AL1978} and Bollob\'{a}s and Cockayne \cite{BoCo1979} independently proved that $\gamma(G) < 2ir(G)$, which can be written $\iota_1(G) < 2ir(G)$, for any graph $G$. In this paper, for a graph $G$ with maximum degree $\Delta$, we establish sharp upper bounds on $\iota_{k}(G)$ in terms of $ir(G)$ for $\Delta - 2 \leq k \leq \Delta + 1$