Some Bounds on the Double Domination of Signed Generalized Petersen Graphs and Signed I-Graphs

In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $\Sigma = (G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset $D \subseteq V(\Sigma)$ is a double dominating set of $\Sigma$ if it satisfies the following conditions: (i) $D$ is a double dominating set of $G$, and (ii) $\Sigma[D:V \setminus D]$ is balanced, where $\Sigma[D:V \setminus D]$ is the subgraph of $\Sigma$ induced by the edges of $\Sigma$ with one end point in $D$ and the other end point in $V \setminus D$. The cardinality of a minimum double dominating set of $\Sigma$ is the double domination number $\gamma_{\times 2}(\Sigma)$. In this paper, we give bounds for the double domination number of signed cubic graphs. We also obtain some bounds on the double domination number of signed generalized Petersen graphs and signed I-graphs.Comment: 13 page

The domination number of on-line social networks and random geometric graphs

We consider the domination number for on-line social networks, both in a stochastic network model, and for real-world, networked data. Asymptotic sublinear bounds are rigorously derived for the domination number of graphs generated by the memoryless geometric protean random graph model. We establish sublinear bounds for the domination number of graphs in the Facebook 100 data set, and these bounds are well-correlated with those predicted by the stochastic model. In addition, we derive the asymptotic value of the domination number in classical random geometric graphs

Degree Sequence Index Strategy

We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the $k$-independence and the $k$-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and $K_{1,r}$-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester