Bounds for the mm-Eternal Domination Number of a Graph

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The $m$-eternaldomination number, \edom(G), of a graph $G$ is the minimum number of guards needed to defend $G$ against any such sequence. We show that if $G$ is a connected graph with minimum degree at least~$2$ and of order~$n \ge 5$, then \edom(G) \le \left\lfloor \frac{n-1}{2} \right\rfloor, and this bound is tight. We also prove that if $G$ is a cubic bipartite graph of order~$n$, then \edom(G) \le \frac{7n}{16}

Perfect Roman domination in regular graphs

A perfect Roman dominating function on a graph G is a function f: V (G) → (0, 1, 2) satisfying the condition that every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted γ Rp(G), is the minimum weight of a perfect Roman dominating function in G. We show that if G is a cubic graph on n vertices, then γRp (G) ≤ 3/4n, and this bound is best possible. Further, we show that if G is a k-regular graph on n vertices with k at least 4, then γ Rp(G) ≤ (k2+k+3/k2+3k+1) n

Packing in regular graphs

A set S of vertices in a graph G is a packing if the vertices in S are pairwise at distance at least 3 apart in G. The packing number of G, denoted by p(G), is the maximum cardinality of a packing in G. Favaron [Discrete Math. 158 (1996), 287–293] showed that if G is a connected cubic graph of order n different from the Petersen graph, then p(G) ≥ n/8. In this paper, we generalize Favaron’s result. We show that for k ≥ 3, if G is a connected k-regular graph of order n that is not a diameter-2 Moore graph, then p(G) ≥ n/(k2 − 1).Mathematics Subject Classification (2010): 05C65.Keywords: Packing, regular graphs, Moore graph

Trees with large m-eternal domination number

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The m-eternal domination number, γm∞(G), of a graph G is the minimum number of guards needed to defend G against any such sequence. We characterize the class of trees of order n with maximum possible m-eternal domination number, which is ⌈[Formula Presented]⌉

Italian domination in trees

The Roman domination number and Italian domination number (also known as the Roman {2}-domination number) are graph labeling problems in which each vertex is labeled with either 0, 1, or 2. In the Roman domination problem, each vertex labeled 0 must be adjacent to at least one vertex labeled 2. In the Italian domination problem, each vertex labeled 0 must have the labels of the vertices in its closed neighborhood sum to at least two. The Italian domination number, γI(G), of a graph G is the minimum possible sum of such a labeling, where the sum is taken over all the vertices in G. It is known that if T is a tree with at least two vertices, then γ(T)+1≤γI(T)≤2γ(T). In this paper, we characterize the trees T for which γ(T)+1=γI(T), and we characterize the trees T for which γI(T)=2γ(T)

Packing in regular graphs

A set S of vertices in a graph G is a packing if the vertices in S are pairwise at distance at least 3 apart in G. The packing number of G, denoted by ρ(G), is the maximum cardinality of a packing in G. Favaron [Discrete Math. 158 (1996), 287–293] showed that if G is a connected cubic graph of order n different from the Petersen graph, then ρ(G) ≥ n/8. In this paper, we generalize Favaron’s result. We show that for k ≥ 3, if G is a connected k-regular graph of order n that is not a diameter-2 Moore graph, then ρ(G) ≥ n/(k2 − 1)

Bounds for the mm-Eternal Domination Number of a Graph

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The $m$-eternaldomination number, \edom(G), of a graph $G$ is the minimum number of guards needed to defend $G$ against any such sequence. We show that if $G$ is a connected graph with minimum degree at least~$2$ and of order~$n \ge 5$, then \edom(G) \le \left\lfloor \frac{n-1}{2} \right\rfloor, and this bound is tight. We also prove that if $G$ is a cubic bipartite graph of order~$n$, then \edom(G) \le \frac{7n}{16}