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

Dimensi Metrik Graf Blok Bebas Anting

Abstrak. Misalkan G = (V(G),E(G)) adalah graf dengan himpunan titik V(G) dan himpunan garis E(G). Representasi dari v terhadap himpunan titik W = {w1, w2, ;wk} V(G) adalah k-tuple r(v|W) = (d(v,w1), d(v,w2), , d(v,wk)). Himpunan W disebut himpunan resolving dari G jika setiap titik mempunyai representasi yang berbeda terhadap W. Titik potong v di G adalah titik di G dengan sifat jika titik v dihapus maka banyaknya komponen G - v akan lebih besar dari banyaknya komponen G. Sebuah blok dari suatu graf adalah subgraf maksimal tanpa titik potong. Graf G disebut graf blok jika dan hanya jika setiap blok dari graf G adalah graf lengkap. Blok dari graf blok yang diperoleh dengan hanya menghapus satu titik potong dari graf blok disebut dengan blok ujung. Blok ujung yang hanya satu titik disebut dengan anting. Pada makalah ini akan dibahas beberapa sifat himpunan resolving dan nilai dimensi metrik dari graf blok yang tidak memiliki anting

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

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