A Nordhaus–Gaddum bound for Roman domination

Abstract

A Roman dominating function of a graph [Formula: see text] is a labeling [Formula: see text] such that every vertex with label [Formula: see text] has a neighbor with label [Formula: see text]. The Roman domination number, [Formula: see text] of [Formula: see text], is the minimum of [Formula: see text] over such functions. Let [Formula: see text] be an [Formula: see text]-vertex graph. Chambers et al. [E. W. Chambers, B. Kinnersley, N. Prince and D. B. West, External Problems for Roman domination Siam J. Discrete Math. 23 (2009) 1575–1586.] proved that if [Formula: see text] is a connected graph of order [Formula: see text], then [Formula: see text], with equality if and only if [Formula: see text] or [Formula: see text] is [Formula: see text] or [Formula: see text]. In this paper, we construct a specific family of graphs [Formula: see text], and prove that if [Formula: see text] and [Formula: see text], then [Formula: see text], and this bound is sharp.