A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

A Theory of Graphs

Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

Proofs from THE BOOK , by Martin Aigner and Günter M. Ziegler. Pp. 199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag).

A survey of Nordhaus-Gaddum type relations

In 1985, Fink and Jacobson gave a generalization of the concepts of domination and independence in graphs For a positive integer k, a subset S of vertices in a graph G = (V, E) is k-dominating if every vertex of V − S is adjacent to at least k vertices in S The subset S is k-independent if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1 In this paper we survey results on k-domination and k-independence

k -Domination and k -Independence in Graphs: A Survey

On the double Roman domination in graphs

Fair domination in graphs

A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbors in S. The k-domination number $\gamma_k(G)$ is the minimum cardinality of a k-dominating set of G. Different upper bounds on $\gamma_{k}(G)$ are known in terms of the order n and the minimum degree $\delta$ of G. In this self-contained article, we present an Erdos-type result, from which some of these bounds follow. In particular, we improve the bound $\gamma_{k}(G) \le (2k- \delta - 1)n/(2k -\delta)$ for $(\delta +3)/2 \le k \le \delta - 1$, proved by Chen and Zhou in 1998. Furthermore, we characterize the extremal graphs in the inequality $\gamma_{k}(G) \le kn/(k +1)$, if $k \le \delta$, of Cockayne et al. This characterization generalizes that of graphs realizing $\gamma_1(G) = \gamma(G) = n/2$. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 33–40, 2008

On k-domination and minimum degree in graphs

Upper bounds on the k-domination number and the k-Roman domination number

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$ . A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant . The $k$-irredundance number of $G$ , denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemanska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

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Adriana Hansberg

Papers

k -Domination and k -Independence in Graphs: A Survey

Fair domination in graphs

On k-domination and minimum degree in graphs

Upper bounds on the k-domination number and the k-Roman domination number

Distance domination and distance irredundance in graphs