Defending the Roman Empire - A new strategy

Motivated 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=(VE) 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 is an element of V- {u,v}, has no undefended vertex. The weight of f is w(f) = Sigma(vis an element ofV) f(v). The weak Roman domination number, denoted -,(G), is the minimum weight of a WRDF in G. We show that for every graph G, gamma(G) less than or equal to gamma(r)(G) less than or equal to 2gamma(G). We characterize graphs G for which gamma(r)(G) = gamma(G) and we characterize forests G for which gamma(r)(G) = 2gamma(G). (C) 2003 Elsevier Science B.V. All rights reserved.