Defending the Roman Empire - A new strategy

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Abstract

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=(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 γ r(G), is the minimum weight of a WRDF in G. We show that for every graph G, γ(G)γ r(G)2γ(G). We characterize graphs G for which γ r(G)=γ(G) and we characterize forests G for which γ r(G)=2γ(G). © 2003 Elsevier Science B.V. All rights reserved.

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Henning, M. A., & Hedetniemi, S. T. (2003). Defending the Roman Empire - A new strategy. In Discrete Mathematics (Vol. 266, pp. 239–251). https://doi.org/10.1016/S0012-365X(02)00811-7

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