A Roman dominating function (RDF) on a graph G = (V; E) is a function f: V! f0; 1; 2g such that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. An RDF f is called an outer independent Roman dominating function (OIRDF) if the set of vertices assigned a 0 under f is an independent set. The weight of an OIRDF is the sum of its function values over all vertices, and the outer independent Roman domination number oiR(G) is the minimum weight of an OIRDF on G. . .

Let G = (V (G), E(G)) be a graph with the set of vertices V (G) and the set of edges E(G). A subset S of E(G) is called a k-nearly independent edge subset if there are exactly k pairs of elements of S that share a common end. Zk(G) is the number of such subsets. This paper studies Z1. Various properties of Z1 are discussed. We characterize the two n-vertex trees with the smallest Z1, as well as the one with the largest value. A conjecture on the n-vertex tree with the second-largest Z1 is proposed.

An outer-independent double Italian dominating function (OIDIDF) on a graph G with vertex set V (G) is a function f: V (G)! f0; 1; 2; 3g such that if f(v) 2 f0; 1g for a vertex v 2 V (G) then P u2N[v] f(u)  3, and the set fu 2 V (G)jf(u) = 0g is independent. The weight of an OIDIDF f is the value w(f) = P v2V (G) f(v). The minimum weight of an OIDIDF on a graph G is called the outer-independent double Italian domination number oidI (G) of G. We present sharp lower bounds for the outer-independent double Italian domination number of a tree in terms of diameter, vertex covering number and the order of the tree.

Let G be a simple graph. An independent set is a set of pairwise non-adjacent vertices. The number of vertices in a maximum independent set of G is denoted by a (G). In this paper, we characterize graphs G with n vertices and with maximum number of maximum independent sets provided that a (G) £2 or a (G) ³n-3.

A 2-rainbow dominating function on a graph G is a function g that assigns to each vertex a set of colors chosen from the subsets of {1, 2} so that for each vertex with g(v) = 0 we have ,uϵ, N(υ,) g(u) = {1, 2}. The weight of a 2-rainbow dominating function g is the value ω,(g) = ∑,υ, ϵ, v(G) |f(υ, )|. A 2-rainbow dominating function g is an independent 2-rainbow dominating function if no pair of vertices assigned nonempty sets are adjacent. The 2-rainbow domination number ɤ, r2(G) (respectively, the inde-pendent 2-rainbow domination number ir2(G)) is the minimum weight of a 2-rainbow dominating function (respectively, independent 2-rainbow dominating function) on G. We prove that for any tree T of order n ≥, 3, with l leaves and s support vertices, ir2(T) ≤, (14n + ʆ,+ s)=20, thus improving the bound given in [Independent 2-rainbow domination in trees, Asian-Eur. J. Math. 8 (2015) 1550035] under certain conditions.

Precue is one of the effective factors on RT. The results of studies done on this factor have shown the reduction of RT as function of amount of precue information; but the amount of effect of number of precued parameters independent of stimulus-response alternatives has not been defined exactly. The present research was performed in order to test the hypothesis that RT in a force production task is function of number of precued parameters. A mixed 3 factional design was performed on 16 (8 male & 8 female) volunteer, non-athletes, right-handed students ranging in age from 20 to 25. Using "parameter precueing apparatus", subjects performed a total of 2400 trials over 5 sessions on successive days (4 blocks of 120 trials blocks per session) under different 2-choice conditions with 1 or 2 precues. The task required production of defined isometric force (3 or 6 kg) to inside or outside, with right or left upper limb as quickly and accurately as possible after displaying precue followed by stimulus. The subjects' RT in different levels of independent variable were analyzed by using a 3 factor design (number of precued parameters *gender* session) with repeated measures on 2 of the factors (number of precued parameters & session). The results indicated the RT was not function of number of precued parameters in this task (P>0.05). The main effect of gender and interaction of number of precued parameters & gender were not significant (P>0.05); but the main effect of session was significant (P<0.05).

An independent Italian dominating function (IID-function) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) $\sum_{u\in N(v)}f(u)\geq2$ when $f(v)=0$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IID-function is the sum of its function values over all vertices, and the independent Italian domination number $i_{I}(G)$ of $G$ is the minimum weight of an IID-function on $G$. In this paper, we initiate the study of the independent Italian bondage number $b_{iI}(G)$ of a graph $G$ having at least one component of order at least three, defined as the smallest size of a set of edges of $G$ whose removal from $G$ increases $i_{I}(G)$. We show that the decision problem associated with the independent Italian bondage problem is NP-hard for arbitrary graphs. Moreover, various upper bounds on $b_{iI}(G)$ are established as well as exact values on it for some special graphs. In particular, for trees $T$ of order at least three, it is shown that $b_{iI}(T)\leq2$.

Let G be a graph. A 2-rainbow dominating function (or 2-RDF) of G is a function f from V (G) to the set of all subsets of the set f1; 2g such that for a vertex v 2 V (G) with f(v) =; , the condition S u2NG(v) f(u) = f1; 2g is ful lled, where NG(v) is the open neighborhood of v. The weight of 2-RDF f of G is the value! (f): = P v2V (G) jf(v)j. The 2-rainbow domination number of G, denoted by r2(G), is the minimum weight of a 2-RDF of G. A 2-RDF f is called an outer independent 2-rainbow dominating function (or OI2-RDF) of G if the set of all v 2 V (G) with f(v) =; is an independent set. The outer independent 2-rainbow domination number oir2(G) is the minimum weight of an OI2-RDF of G. In this paper, we obtain the outer independent 2-rainbow domination number of Pm  Pn and Pm  Cn. Also we determine the value of oir2(Cm2Cn) when m or n is even.