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

Let G = (V, E) be a graph. A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V −D has a neighbor in V −D. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number of G. In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for...

Let G = (V,E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to at least one vertex in S. A total dominating set S ⊆ V is a paired-dominating set if the induced subgraph G[S] has at least one perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-domination set of G. In this paper, we provide a c...

Recently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary...

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...