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)

Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Roman and inverse roman domination in network of triangles

In graph G (V, E), a function f : V → {0, 1 2} is said to be a Roman Dominating Function (RDF). If ∀u ∈ V, f(u) = 0 is adjacent to at least one vertex v ∈ V such that f(v) = 2. The weight of f is given by w(f) = P v∈V f(v). The Roman Domination Number (RDN) denoted by γR(G) is the minimum weight among all RDF in G. If V −D contains a RDF f 1 : V → {0, 1, 2}, where D is the set of vertices v, f(v) > 0, then f 1 is called Inverse Roman Dominating Function (IRDF) on a graph G with respect to the RDF f. The Inverse Roman Domination Number (IRDN) denoted by γ 1 R(G) is the minimum weight among all IRDF in G. In this paper we find RDN and IRDN of few triangulations graphs.Publisher's Versio