A Roman dominating function (RDF) on a graph $G$ is a function $f : V (G) to {0, 1, 2}$satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least onevertex $v$ for which $f(v) = 2$. A Roman dominating function $f$ is called an outer-independentRoman dominating function (OIRDF) on $G$ if the set ${vin Vmid f(v)=0}$ is independent.The (outer-independent) Roman dom...

A set S of vertices in a graph G is dominating if every vertex not adjacent to S. If, addition, an independent set, then set. The domination number i(G) the minimum cardinality G. In Goddard and Henning (Discrete Math 313:839–854, 2013) conjectured that connected cubic order n, $$i(G) \le \frac{3}{8}n$$ , except complete bipartite $$K_{3,3}$$ or 5-prism $$C_5 \, \Box K_2$$ . Further they constr...

The concepts of covering and matching in fuzzy graphs using strong arcs are introduced and obtained the relationship between them analogous to Gallai’s results in graphs. The notion of paired domination in fuzzy graphs using strong arcs is also studied. The strong paired domination number γspr of complete fuzzy graph and complete bipartite fuzzy graph is determined and obtained bounds for the s...

Abstract A set S of vertices in a graph G is dominating if every vertex not ad jacent to . If, addition, an independent set, then set. The domination number i ( ) the minimum cardinality subdivision $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G<...