Roman domination excellent graphs: trees
Vladimir
Samodivkin
University of Architecture, Civil Еngineering and Geodesy;
Department of Mathematics
author
text
article
2018
eng
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 $gamma(G-v) = gamma (G)$ 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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
1
no.
2018
1
24
http://comb-opt.azaruniv.ac.ir/article_13654_ec8df599ae3874367c247f6fb520698c.pdf
dx.doi.org/10.22049/cco.2017.25806.1041
Product version of reciprocal degree distance of composite graphs
K
Pattabiraman
Annamalai University
author
text
article
2018
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
1
no.
2018
25
35
http://comb-opt.azaruniv.ac.ir/article_13655_225b463f2a3ca5e41aee2b3b437d11c2.pdf
dx.doi.org/10.22049/cco.2017.26018.1067
Total $k$-Rainbow domination numbers in graphs
Hossein
Abdollahzadeh Ahangar
Babol Noshirvani University of Technology
author
Jafar
Amjadi
Azarbaijan Shahid Madani University
author
Nader
Jafari Rad
Shahrood University of Technology
author
Vladimir
D. Samodivkin
University of Architecture, Civil Engineering and Geodesy
author
text
article
2018
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
1
no.
2018
37
50
http://comb-opt.azaruniv.ac.ir/article_13683_b5784dd717acd4308580ca847ce38c2b.pdf
dx.doi.org/10.22049/cco.2018.25719.1021
An infeasible interior-point method for the $P*$-matrix linear complementarity problem based on a trigonometric kernel function with full-Newton step
Behrouz
Kheirfam
Azarbaijan Shahid Madani University
author
Masoumeh
Haghighi
Azarbaijan Shahid Madani University
author
text
article
2018
eng
An infeasible interior-point 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 interior-point methods for$P_*$-matrix linear complementarity problem.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
1
no.
2018
51
70
http://comb-opt.azaruniv.ac.ir/article_13693_2409f47f2535c47bbf7f6f1c4e57f291.pdf
dx.doi.org/10.22049/cco.2018.25801.1038
Double Roman domination and domatic numbers of graphs
Lutz
Volkmann
RWTH Aachen University
author
text
article
2018
eng
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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
1
no.
2018
71
77
http://comb-opt.azaruniv.ac.ir/article_13744_0b21f0f7d95e9ab99c3422cf6f3acc77.pdf
dx.doi.org/10.22049/cco.2018.26125.1078
Mixed Roman domination and 2-independence in trees
Nasrin
Dehgardi
Sirjan University of Technology, Sirjan 78137, Iran
author
text
article
2018
eng
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 2-independent set of $G$ if every vertex of $S$ has at most one neighbor in $S$. The minimum cardinality of a 2-independent set of $G$ is the 2-independence 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.
Communications in Combinatorics and Optimization
Azarbaijan Shahid Madani University
2538-2128
3
v.
1
no.
2018
79
91
http://comb-opt.azaruniv.ac.ir/article_13747_2ec084aafd9f82151a96717d372ecd5b.pdf
dx.doi.org/10.22049/cco.2018.25964.1062