ROMAN DOMINATION ON 2-CONNECTED GRAPHS

A Roman dominating function of a graph G is a function f: V(G) -> {0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u) = 2. The weight of f is w(f) = Sigma(v is an element of V(G)) f(v). The Roman domination number gamma(R)(G) of G is the minimum weight of a Roman dominating function of G. Chambers, Kinnersley, Prince, and West [SIAM J. Discrete Math., 23 (2009), pp. 1575-1586] conjectured that gamma(R)(G) <= inverted right perpendicular2n/3inverted left perpendicular for any 2-connected graph G of n vertices. This paper gives counterexamples to the conjecture and proves that gamma(R)(G) <= max{inverted right perpendicular2n/3inverted left perpendicular, 23n/34} for any 2-connected graph G of n vertices. We also characterize 2-connected graphs G for which gamma(R)(G) = 23n/34 when 23n/34 > inverted right perpendicular2n/3inverted left perpendicular.