A k-dominating set of a graph G is a set S of vertices of G such that every vertex outside of S has k neighbors in S. The k-domination number of G, written γk(G), is the size of the smallest k-dominating set in G. In this paper, we derive sharp upper and lower bounds on γk(G) + γk(G) and γk(G)γk(G), where G is the complement of G. We use the results for k = 2 to prove a conjecture of Alon, Balo...

In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select dominating set and Staller's claim closed neighborhood of some vertex. We study cases when Staller can win game. If Dominator (resp., Staller) starts game, then $\gamma_{\rm SMB}(G)$ SMB}'(G)$) denotes minimum number moves needs win. For every positive integer $k$, trees $T$ with SMB}'(T)=k$ are characteri...

We consider settings in which voters vote in sequence, each voter knows the votes of the earlier voters and the preferences of the later voters, and voters are strategic. This can be modeled as an extensive-form game of perfect information, which we call a Stackelberg voting game. We first propose a dynamic-programming algorithm for finding the backward-induction outcome for any Stackelberg vot...

A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices inducing a graph with a perfect matching. The paired-domination number of G, denoted by γpr (G), is the minimum cardinality of a paired-dominating set of G. We consider graphs of order n ≥ 6, minimum degree δ such that G and G do not have an isolated vertex and we prove that – if γpr (G) > 4 an...

Suppose G = (V , E) is a simple graph and k is a fixed positive integer. A subset D ⊆ V is a distance k-dominating set of G if for every u ∈ V , there exists a vertex v ∈ D such that dG(u, v) ≤ k, where dG(u, v) is the distance between u and v in G. A set D ⊆ V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph G[D] contains a perfect matchin...