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...

We initiate the study of total outer-independent domination in graphs. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \ D is independent. The total outer-independent domination number of a graph G is the minimum cardinality of a total outer-independent dominating set of G. First we discuss the ...

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...

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<...

A subset D of ( ) V G is called an equitable dominating set if for every ( ) v V G D   there exists a vertex u D  such that ( ) uv E G  and | ( ) ( ) | 1 deg u deg v   . A subset D of ( ) V G is called an equitable independent set if for any , u D v   ( ) e N u for all { } v D u   . The concept of equi independent equitable domination is a combination of these two important concepts. ...