90 research outputs found
Upper domination and upper irredundance perfect graphs
Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) = IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result. © 1998 Elsevier Science B.V. All rights reserved
A semi-induced subgraph characterization of upper domination perfect graphs
Let β(G) and Γ(G) be the independence number and the upper domination number of a graph G, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. The class of Γ-perfect graphs generalizes such well-known classes of graphs as strongly perfect graphs, absorbantly perfect graphs, and circular arc graphs. In this article, we present a characterization of Γ-perfect graphs in terms of forbidden semi-induced subgraphs. Key roles in the characterization are played by the odd prism and the even Möbius ladder, where the prism and the Möbius ladder are well-known 3-regular graphs [2]. Using the semi-induced subgraph characterization, we obtain a characterization of K 1.3-free Γ-perfect graphs in terms of forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc
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 αrγs(k)-perfect graphs
AbstractFor some integer k⩾0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)⩽k for every induced subgraph H of G. For r⩾1 let αr and γr denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α1γ1(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study αrγs(k)-perfect graphs for r,s⩾1. We prove several properties of minimal αrγs(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α2r−1γr(k)-perfect graphs for r⩾1. Furthermore, we characterize claw-free α2γ2(k)-perfect graphs
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
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
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