2018
3
1
0
91
Roman domination excellent graphs: trees
2
2
A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$function.A graph G is said to be $gamma_R$excellent if for each vertex $x in V$there is a $gamma_R$function $h_x$ on $G$ with $h_x(x) not = 0$. We present a constructive characterization of $gamma_R$excellent trees using labelings. A graph $G$ is said to be in class $UVR$ if $gamma(Gv) = gamma (G)$ for each $v in V$, where $gamma(G)$ is the domination number of $G$. We show that each tree in $UVR$ is $gamma_R$excellent.
1

1
24


Vladimir
Samodivkin
University of Architecture, Civil Еngineering and Geodesy;
Department of Mathematics
University of Architecture, Civil Еngineering
Bulgaria
vl.samodivkin@gmail.com
Roman domination number
excellent graphs
trees
Product version of reciprocal degree distance of composite graphs
2
2
A {it topological index} of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. In this paper, we present the upper bounds for the product version of reciprocal degree distance of the tensor product, join and strong product of two graphs in terms of other graph invariants including the Harary index and Zagreb indices.
1

25
35


K
Pattabiraman
Annamalai University
Annamalai University
India
pramank@gmail.com
Degree distance
reciprocal degree distance
composite graph
Total $k$Rainbow domination numbers in graphs
2
2
Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$rainbow dominating function} (or a {it $k$RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} of a $k$RDF $f$ of$G$ is the value $omega (f)=sum _{vin V(G)}f(v)$. A $k$rainbowdominating function $f$ in a graph with no isolated vertex is calleda {em total $k$rainbow dominating function} if the subgraph of $G$induced by the set ${v in V(G) mid f (v) not = {color{blue}emptyset}}$ has no isolated vertices. The {em total $k$rainbow domination number} of $G$, denoted by$gamma_{trk}(G)$, is the minimum weight of a total $k$rainbowdominating function on $G$. The total $1$rainbow domination is thesame as the total domination. In this paper we initiate thestudy of total $k$rainbow domination number and we investigate itsbasic properties. In particular, we present some sharp bounds on thetotal $k$rainbow domination number and we determine {color{blue}the} total$k$rainbow domination number of some classes of graphs.
1

37
50


Hossein
Abdollahzadeh Ahangar
Babol Noshirvani University of Technology
Babol Noshirvani University of Technology
Iran
ha.ahangar@yahoo.com


Jafar
Amjadi
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
Iran
jamjadi@azaruniv.ac.ir


Nader
Jafari Rad
Shahrood University of Technology
Shahrood University of Technology
Iran
n.jafarirad@gmail.com


Vladimir
D. Samodivkin
University of Architecture, Civil Engineering and Geodesy
University of Architecture, Civil Engineering
Bulgaria
vlsam_fte@uacg.bg
$k$rainbow dominating function
$k$rainbow domination number
total $k$rainbow dominating function
total $k$rainbow domination number
An infeasible interiorpoint method for the $P*$matrix linear complementarity problem based on a trigonometric kernel function with fullNewton step
2
2
An infeasible interiorpoint algorithm for solving the$P_*$matrix linear complementarity problem based on a kernelfunction with trigonometric barrier term is analyzed. Each (main)iteration of the algorithm consists of a feasibility step andseveral centrality steps, whose feasibility step is induced by atrigonometric kernel function. The complexity result coincides withthe best result for infeasible interiorpoint methods for$P_*$matrix linear complementarity problem.
1

51
70


Behrouz
Kheirfam
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
Iran
b.kheirfam@azaruniv.edu


Masoumeh
Haghighi
Azarbaijan Shahid Madani University
Azarbaijan Shahid Madani University
Iran
b.kheirfam@yahoo.com
Linear complementarity problem
FullNewton step
Infeasible interiorpoint method
Kernel function
Polynomial complexity
Double Roman domination and domatic numbers of graphs
2
2
A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function$f:V(G)rightarrow{0,1,2,3}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least twoneighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must haveat least one neighbor $u$ with $f(u)ge 2$. The weight of a double Roman dominating function $f$ is the sum$sum_{vin V(G)}f(v)$, and the minimum weight of a double Roman dominating function on $G$ is the double Romandomination number $gamma_{dR}(G)$ of $G$.A set ${f_1,f_2,ldots,f_d}$ of distinct double Roman dominating functions on $G$ with the property that$sum_{i=1}^df_i(v)le 3$ for each $vin V(G)$ is called in cite{v} a double Roman dominating family (of functions)on $G$. The maximum number of functions in a double Roman dominating family on $G$ is the double Roman domatic numberof $G$.In this note we continue the study the double Roman domination and domatic numbers. In particular, we presenta sharp lower bound on $gamma_{dR}(G)$, and we determine the double Roman domination and domatic numbers of someclasses of graphs.
1

71
77


Lutz
Volkmann
RWTH Aachen University
RWTH Aachen University
Germany
volkm@math2.rwthaachen.de
domination
Double Roman domination number
Double Roman domatic number
Mixed Roman domination and 2independence in trees
2
2
Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacentor incident to at least one element $yin Vcup E$ for which $f(y)=2$. The weight of anMRDF $f$ is $sum _{xin Vcup E} f(x)$. The mixed Roman domination number $gamma^*_R(G)$ of $G$ isthe minimum weight among all mixed Roman dominating functions of $G$. A subset $S$ of $V$ is a 2independent set of $G$ if every vertex of $S$ has at most one neighbor in $S$. The minimum cardinality of a 2independent set of $G$ is the 2independence number $beta_2(G)$. These two parameters are incomparable in general, however, we show that if $T$ is a tree, then $frac{4}{3}beta_2(T)ge gamma^*_R(T)$ and we characterize all trees attaining the equality.
1

79
91


Nasrin
Dehgardi
Sirjan University of Technology, Sirjan 78137, Iran
Sirjan University of Technology, Sirjan 78137,
Iran
ndehgardi@gmail.com
Mixed Roman dominating function
Mixed Roman domination number
2independent set
2independence number