On global location-domination in graphs

A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the location-domination number $\lambda(G)$. An LD-set $S$ of a graph $G$ is global if it is an LD-set of both $G$ and its complement $\overline{G}$. The global location-domination number $\lambda_g(G)$ is the minimum cardinality of a global LD-set of $G$. In this work, we give some relations between locating-dominating sets and the location-domination number in a graph and its complement.Comment: 15 pages: 2 tables; 8 figures; 20 reference

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

On global location-domination in graphs

A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number lambda(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G'. The global location-domination number lambda g(G) is introduced as the minimum cardinality of a global LD-set of G. In this paper, some general relations between LD-codes and the location-domination number in a graph and its complement are presented first. Next, a number of basic properties involving the global location-domination number are showed. Finally, both parameters are studied in-depth for the family of block-cactus graphs.Postprint (published version

Global location-domination in graphs

Domination, Global domination, Locating domination, Complement graph, Block-cactus, TreesA dominating set S of a graph G is called locating-dominating, LD-setfor short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number (G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G. The global location-domination number g(G) is the minimum cardinality of a global LD-set of G. In this work,we give some relations between locating-dominating sets and the location-domination number in a graph and its complementPreprin