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 Communications in Combinatorics and OptimizationNumber of Followers: 0     Open Access journal ISSN (Print) 2538-2128 - ISSN (Online) 2538-2136 Published by Azarbaijan Shahide Madani University  [1 journal]
• Complexity and approximation ratio of semitotal domination in graphs

• Abstract: A set \$S subseteq V(G)\$ is a semitotal dominating set of a graph \$G\$ if it is a dominating set of \$G\$ andevery vertex in \$S\$ is within distance 2 of another vertex of \$S\$. Thesemitotal domination number \$gamma_{t2}(G)\$ is the minimumcardinality of a semitotal dominating set of \$G\$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal domination problem in any graph of maximum degree \$Delta\$ can be approximated with an approximationratio of \$2+ln(Delta-1)\$.
PubDate: Mon, 04 Jun 2018 19:30:00 +010

• Roman domination excellent graphs: trees

• Abstract: 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.
PubDate: Thu, 31 May 2018 19:30:00 +010

• Product version of reciprocal degree distance of composite graphs

• Abstract: 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.
PubDate: Thu, 31 May 2018 19:30:00 +010

• Total \$k\$-Rainbow domination numbers in graphs

• Abstract: 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.
PubDate: Thu, 31 May 2018 19:30:00 +010

• An infeasible interior-point method for the \$P*\$-matrix linear
complementarity problem based on ...

• Abstract: 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.
PubDate: Thu, 31 May 2018 19:30:00 +010

• Double Roman domination and domatic numbers of graphs

• Abstract: 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.
PubDate: Thu, 31 May 2018 19:30:00 +010

• Mixed Roman domination and 2-independence in trees

• Abstract: ‎‎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.
PubDate: Thu, 31 May 2018 19:30:00 +010

• On the harmonic index of bicyclic graphs

• Abstract: The harmonic index of a graph \$G\$, denoted by \$H(G)\$, is defined asthe sum of weights \$2/[d(u)+d(v)]\$ over all edges \$uv\$ of \$G\$, where\$d(u)\$ denotes the degree of a vertex \$u\$. Hu and Zhou [Y. Hu and X. Zhou, WSEAS Trans. Math. {bf 12} (2013) 716--726] proved that for any bicyclic graph \$G\$ of order \$ngeq 4\$, \$H(G)le frac{n}{2}-frac{1}{15}\$ and characterize all extremal bicyclic graphs.In this paper, we prove that for any bicyclic graph \$G\$ of order \$ngeq 4\$ and maximum degree \$Delta\$, \$\$frac{1}{2} H(G)le left{begin{array}{ll}frac{3n-1}{6} & {rm if}; Delta=4&frac{2Delta-n-3}{Delta+1}+frac{n-Delta+3}{Delta+2}+frac{1}{2}+frac{n-Delta-1}{3} & {rm if};Deltage 5 ;{rm and}; nle 2Delta-4&frac{Delta}{Delta+2}+frac{Delta-4}{3}+frac{n-2Delta+4}{4} & {rm if};Deltage 5 ;{rm and};nge 2Delta-3,end{array}right.\$\$and characterize all extreme bicyclic graphs.
PubDate: Sun, 22 Apr 2018 19:30:00 +010

• Classification of rings with toroidal annihilating-ideal graph

• Abstract: Let R be a non-domain commutative ring with identity and A(R) be theset of non-zero ideals with non-zero annihilators. We call an ideal I of R, anannihilating-ideal if there exists a non-zero ideal J of R such that IJ = (0).The annihilating-ideal graph of R is defined as the graph AG(R) with the vertexset A(R) and two distinct vertices I and J are adjacent if and only if IJ =(0). In this paper, we characterize all commutative Artinian nonlocal rings Rfor which AG(R) has genus one.
PubDate: Thu, 19 Apr 2018 19:30:00 +010

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