Domination theory is a well-established topic in graph theory, as well one of the most active research areas. Interest this area partly explained by its diversity applications to real-world problems, such facility location computer and social networks, monitoring communication, coding algorithm design, among others. In last two decades, functions defined on graphs have attracted attention sever...

We analyze the graph-theoretic formalization of Roman domination, dating back to the military strategy of Emperor Constantine, from a parameterized perspective. More specifically, we prove that this problem is W[2]-complete for general graphs. However, parameterized algorithms are presented for graphs of bounded treewidth and for planar graphs. Moreover, it is shown that a parametric dual of Ro...

In this work, we study the signed Roman domination number of the join of graphs. Specially, we determine it for the join of cycles, wheels, fans, and friendship graphs.

An outer-independent double Roman dominating function (OIDRDF) of a graph G is h:V(G)→{0,1,2,3}</...

We provide two algorithms counting the number of minimum Roman dominating functions of a graph on n vertices in (1.5673) n time and polynomial space. We also show that the time complexity can be reduced to (1.5014) n if exponential space is used. Our result is obtained by transforming the Roman domination problem into other combinatorial problems on graphs for which exact algorithms already exist.

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is double Roman dominating of if for each v ∈ with f(v) = 0, has two adjacent vertices u and w which f(u) f(w) 2 or an vertex 3, to either 3. The minimum weight ωG(f) P v∈V the domination number G. In this paper, we continue study introduced studied by R.A. Beeler et al. in [2]. First, characterize some numbers small values terms 2...