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

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

Contributions to the theory of graphic sequences

In this article we present a new version of the Erdős-Gallai theorem concerning graphicness of the degree sequences. The best conditions of all known on the reduction of the number of Erdős-Gallai inequalities are given. Moreover, we prove a criterion of the bipartite graphicness and give a sufficient condition for a sequence to be graphic which does not require checking of any Erdős-Gallai inequality. © 1992

Disproof of a Conjecture in the Domination Theory

In [1] C. Barefoot, F. Harary and K. Jones conjectured that for cubic graphs with connectivity three the difference between the domination and independent domination numbers is at most one. We disprove this conjecture and give an exhaustive answer to the question: “What is the difference between the domination and independent domination numbers for cubic graphs with given connectivity?” © 1994, Springer-Verlag. All rights reserved

Locally Well-Dominated and Locally Independent Well-Dominated Graphs

In this article we present characterizations of locally well-dominated graphs and locally independent well-dominated graphs, and a sufficient condition for a graph to be k-locally independent well-dominated. Using these results we show that the irredundance number, the domination number and the independent domination number can be computed in polynomial time within several classes of graphs, e.g., the class of locally well-dominated graphs