Defending the Roman Empire—A new strategy

AbstractMotivated by an article by Ian Stewart (Defend the Roman Empire!, Scientific American, Dec. 1999, pp. 136–138), we explore a new strategy of defending the Roman Empire that has the potential of saving the Emperor Constantine the Great substantial costs of maintaining legions, while still defending the Roman Empire. In graph theoretic terminology, let G=(V,E) be a graph and let f be a function f:V→{0,1,2}. A vertex u with f(u)=0 is said to be undefended with respect to f if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u)=0 is adjacent to a vertex v with f(v)>0 such that the function f′:V→{0,1,2}, defined by f′(u)=1, f′(v)=f(v)−1 and f′(w)=f(w) if w∈V−{u,v}, has no undefended vertex. The weight of f is w(f)=∑v∈Vf(v). The weak Roman domination number, denoted γr(G), is the minimum weight of a WRDF in G. We show that for every graph G, γ(G)⩽γr(G)⩽2γ(G). We characterize graphs G for which γr(G)=γ(G) and we characterize forests G for which γr(G)=2γ(G)

Vertex Sequences in Graphs

We consider a variety of types of vertex sequences, which are defined in terms of a requirement that the next vertex in the sequence must meet. For example, let S = (v1, v2, …, vk ) be a sequence of distinct vertices in a graph G such that every vertex vi in S dominates at least one vertex in V that is not dominated by any of the vertices preceding it in the sequence S. Such a sequence of maximal length is called a dominating sequence since the set {v1, v2, …, vk } must be a dominating set of G. In this paper we survey the literature on dominating and other related sequences, and propose for future study several new types of vertex sequences, which suggest the beginning of a theory of vertex sequences in graphs

The Upper Domatic Number of a Graph

Let (Formula presented.) be a graph. For two disjoint sets of vertices (Formula presented.) and (Formula presented.), set (Formula presented.) dominates set (Formula presented.) if every vertex in (Formula presented.) is adjacent to at least one vertex in (Formula presented.). In this paper we introduce the upper domatic number (Formula presented.), which equals the maximum order (Formula presented.) of a vertex partition (Formula presented.) such that for every (Formula presented.), (Formula presented.), either (Formula presented.) dominates (Formula presented.) or (Formula presented.) dominates (Formula presented.), or both. We study properties of the upper domatic number of a graph, determine bounds on (Formula presented.), and compare (Formula presented.) to a related parameter, the transitivity (Formula presented.) of (Formula presented.)

Client–Server and Cost Effective Sets in Graphs

For any integer k≥0, a set of vertices S of a graph G=(V,E) is k-cost-effective if for every v∈S,|N(v)∩(V∖S)|≥|N(v)∩S|+k. In this paper we study the minimum cardinality of a maximal k-cost-effective set and the maximum cardinality of a k-cost-effective set. We obtain Gallai-type results involving the k-cost-effective and global k-offensive alliance parameters, and we provide bounds on the maximum k-cost-effective number. Finally, we consider k-cost-effective sets that are also dominating. We show that computing the k-cost-effective domination number is NP-complete for bipartite graphs. Moreover, we note that not all trees have a k-cost-effective dominating set and give a constructive characterization of those that do

Client–server and cost effective sets in graphs

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On perfect neighborhood sets in graphs

AbstractLet G = (V, E) be a graph and let S ⊆ V.. The set S is a dominating set of G is every vertex of V − S is adjacent to a vertex of S. A vertex v of G is called S-perfect if |N[ν]∩S| = 1 where N[v] denotes the closed neighborhood of v. The set S is defined to be a perfect neighborhood set of G if every vertex of G is S-perfect or adjacent with an S-perfect vertex. We prove that for all graphs G, Θ(G) = Γ(G) where Γ(G) is the maximum cardinality of a minimal dominating set of G and where Θ(G) is the maximum cardinality among all perfect neighborhood sets of G

Iterated colorings of graphs

AbstractFor a graph property P, in particular maximal independence, minimal domination and maximal irredundance, we introduce iterated P-colorings of graphs. The six graph parameters arising from either maximizing or minimizing the number of colors used for each property, are related by an inequality chain, and in this paper we initiate the study of these parameters. We relate them to other well-studied parameters like chromatic number, give alternative characterizations, find graph classes where they differ by an arbitrary amount, investigate their monotonicity properties, and look at algorithmic issues

k-Efficient Partitions of Graphs

A set S = {u1, u2,..., ut} of vertices of G is an efficient dominating set if every vertex of G is dominated exactly once by the vertices of S. Letting Ui denote the set of vertices dominated by ui, we note that {U1, U2,... Ut} is a partition of the vertex set of G and that each Ui contains the vertex ui and all the vertices at distance 1 from it in G. In this paper, we generalize the concept of efficient domination by considering k-efficient domination partitions of the vertex set of G, where each element of the partition is a set consisting of a vertex ui and all the vertices at distance di from it, where di ∈ {0, 1,..., k}. For any integer k ≥ 0, the k-efficient domination number of G equals the minimum order of a k-efficient partition of G. We determine bounds on the k-efficient domination number for general graphs, and for k ∈ {1, 2}, we give exact values for some graph families. Complexity results are also obtained