For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v|W ) = (d(v,w1), d(v,w2), . . . , d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W . A resolv...

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$....

Article history: Received 14 November 2008 Received in revised form 15 January 2012 Accepted 27 February 2012 Available online xxxx

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. After each attack, if the guards can “move” to form a dominating set that contains the attacked vertex, then the guards have successfully defended against the attack. We wish to determine the minimum number of guards required to successfully defend against any possible sequence...