Laplacian Distribution and Domination

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\frac{\gamma(G)}{m_G[0,1)} \not\in O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$

The Ratio of the Irredundance Number and the Domination Number for Block-Cactus Graphs

Let γ(G) and ir(G) denote the domination number and the irredundance number of a graph G, respectively. Allan and Laskar [Proc. 9th Southeast Conf. on Combin., Graph Theory & Comp. (1978) 43-56] and Bollobás and Cockayne [J. Graph Theory (1979) 241-249] proved independently that γ(G) < 2ir(G) for any graph G. For a tree T, Damaschke [Discrete Math. (1991) 101-104] obtained the sharper estimation 2γ(T) < 3ir(T). Extending Damaschke's result, Volkmann [Discrete Math. (1998) 221-228] proved that 2γ(G) ≤ 3ir(G) for any block graph G and for any graph G with cyclomatic number μ(G) ≤ 2. Volkmann also conjectured that 5γ(G) < 8ir(G) for any cactus graph. In this article we show that if G is a block-cactus graph having π(G) induced cycles of length 2 (mod 4), then γ(G)(5π(G) + 4) ≤ ir(G)(8π(G) + 6). This result implies the inequality 5γ(G) ≤ 8ir(G) for a block-cactus graph G, thus proving the above conjecture. © 1998 John Wiley & Sons, Inc

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

A Greedy Partition Lemma for Directed Domination

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $\gamma(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $\Gamma_d(G)$, which is the maximum directed domination number $\gamma(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if $\alpha$ denotes the independence number of a graph $G$, we show that $\alpha \le \Gamma_d(G) \le \alpha(1+2\ln(n/\alpha))$.Comment: 12 page