A characterization of trees with equal 2-domination and 2-independence numbers

A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $\gamma_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $\alpha_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.Comment: 17 pages, 4 figure

Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.Peer ReviewedPostprint (author's final draft

Maker-Breaker domination number

The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $\gamma_{{\rm MB}}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $\gamma_{{\rm MB}}'(G)$. Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant $\gamma_{{\rm MB}}(G)$ is also compared with the domination number. Using the Erd\H{o}s-Selfridge Criterion a large class of graphs $G$ is found for which $\gamma_{{\rm MB}}(G) > \gamma(G)$ holds. Residual graphs are introduced and used to bound/determine $\gamma_{{\rm MB}}(G)$ and $\gamma_{{\rm MB}}'(G)$. Using residual graphs, $\gamma_{{\rm MB}}(T)$ and $\gamma_{{\rm MB}}'(T)$ are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.Comment: 20 pages, 5 figure