33 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
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
The domination parameters of cubic graphs
Let ir(G), γ(G), i(G), β0(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any nonnegative integers k 1, k 2, k 3, k 4, k 5 there exists a cubic graph G satisfying the following conditions: γ(G) - ir(G) ≤ k 1, i(G) - γ(G) ≤ k 2, β0(G) - i(G) > k 3, Γ(G) - β0(G) - k 4, and IR(G) - Γ(G) - k 5. This result settles a problem posed in [9]. © Springer-Verlag 2005
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
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
On the Domination Chain of m by n Chess Graphs
A survey of the six domination chain parameters for both square and rectangular chess boards are discussed
Strong Domination Index in Fuzzy Graphs
Topological indices play a vital role in the area of graph theory and fuzzy
graph (FG) theory. It has wide applications in the areas such as chemical graph
theory, mathematical chemistry, etc. Topological indices produce a numerical
parameter associated with a graph. Numerous topological indices are studied due
to its applications in various fields. In this article a novel idea of
domination index in a FG is defined using weight of strong edges. The strong
domination degree (SDD) of a vertex u is defined using the weight of minimal
strong dominating set (MSDS) containing u. Idea of upper strong domination
number, strong irredundance number, strong upper irredundance number, strong
independent domination number, and strong independence number are explained and
illustrated subsequently. Strong domination index (SDI) of a FG is defined
using the SDD of each vertex. The concept is applied on various FGs like
complete FG, complete bipartite and r-partite FG, fuzzy tree, fuzzy cycle and
fuzzy stars. Inequalities involving the SDD and SDI are obtained. The union and
join of FG is also considered in the study. Applications for SDD of a vertex is
provided in later sections. An algorithm to obtain a MSDS containing a
particular vertex is also discussed in the article
Indicated domination game
Motivated by the success of domination games and by a variation of the
coloring game called the indicated coloring game, we introduce a version of
domination games called the indicated domination game. It is played on an
arbitrary graph by two players, Dominator and Staller, where Dominator
wants to finish the game in as few rounds as possible while Staller wants just
the opposite. In each round, Dominator indicates a vertex of that has
not been dominated by previous selections of Staller, which, by the rules of
the game, forces Staller to select a vertex in the closed neighborhood of .
The game is finished when all vertices of become dominated by the vertices
selected by Staller. Assuming that both players are playing optimally according
to their goals, the number of selected vertices during the game is the
indicated domination number, , of .
We prove several bounds on the indicated domination number expressed in terms
of other graph invariants. In particular, we find a place of the new graph
invariant in the well-known domination chain, by showing that for all graphs , and by showing that the indicated domination
number is incomparable with the game domination number and also with the upper
irredundance number. In connection with the trivial upper bound , we characterize the class of graphs attaining the bound
provided that . We prove that in trees, split graphs and
grids the indicated domination number equals the independence number. We also
find a formula for the indicated domination number of powers of paths, from
which we derive that there exist graphs in which the indicated domination
number is arbitrarily larger than the upper irredundance number.Comment: 19 page