ErdH{o}s [On Sch"utte problem, Math. Gaz. 47 (1963)] proved that every tournament on $n$ vertices has a directed dominating set of at most $log (n+1)$ vertices, where $log$ is the logarithm to base $2$. He also showed that there is a tournament on $n$ vertices with no directed domination set of cardinality less than $log n - 2 log log n + 1$. This notion of directed domination number has been g...

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...

A set S ⊆ V of vertices in a graph G = (V,E) is called a dominating set if every vertex in V − S is adjacent to at least one vertex in S. Domination in graphs is a well-studied branch of graph theory, and is the subject of two books by Haynes, Hedetniemi and Slater [8, 9]. However, about 90% of the papers on domination have considered only undirected graphs. Thus, relatively little is known abo...

In this research study, we introduce the concept of bipolar neutrosophic graphs. We present the dominating and independent sets of bipolar neutrosophic graphs. We describe novel multiple criteria decision making methods based on bipolar neutrosophic sets and bipolar neutrosophic graphs. We also develop an algorithm for computing domination in bipolar neutrosophic graphs. Key-words: Bipolar neut...

For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset D ⊆ V (G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies k-tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for...

Given a graph \(G=(V(G), E(G))\), the size of minimum dominating set, paired and total set G are denoted by \(\gamma (G)\), _{pr}(G)\), _{t}(G)\), respectively. For positive integer k, k-packing in is \(S \subseteq V(G)\) such that for every pair distinct vertices u v S, distance between at least \(k+1\). The number order largest \(\rho _{k}(G)\). It well known _{pr}(G) \le 2\gamma (G)\). In th...

The ratio of the connected domination number, γc, and the domination number, γ, is strictly bounded from above by 3. It was shown by Zverovich that for every connected (P5, C5)-free graph, γc = γ. In this paper, we investigate the interdependence of γ and γc in the class of (Pk, Ck)-free graphs, for k ≥ 6. We prove that for every connected (P6, C6)-free graph, γc ≤ γ+1 holds, and there is a fam...