Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Signed total Italian k-domination in graphs1711831411210.22049/cco.2020.26919.1164ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20200819Let <em>k ≥ 1</em> be an integer, and let <em>G</em> be a finite and simple graph with vertex set <em>V (G)</em>. <br />A signed total Italian <em>k</em>-dominating function (STIkDF) on a graph <em>G</em> is a function<br /><em>f : V (G) → {−1, 1, 2}</em> satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex<br /> <em>v ∈ V (G)</em>, where <em>N(v)</em> is the neighborhood of $v$, and each vertex <em>u</em> with <em>f(u)=-1</em> is adjacent to<br /> a vertex <em>v</em> with <em>f(v)=2</em> or to two vertices <em>w</em> and <em>z</em> with <em>f(w)=f(z)=1</em>. The weight of an STIkDF <em>f</em> is<br />$omega(f)=sum_{vin V(G)}f(v)$. The signed total Italian <em>k</em>-domination number $gamma_{stI}^k(G)$ of <em>G</em> is the<br /> minimum weight of an STIkDF on <em>G</em>. In this paper we initiate the study of the signed total Italian <em>k</em>-domination<br />number of graphs, and we present different bounds on $gamma_{stI}^k(G)$. In addition, we determine the<br />signed total Italian <em>k</em>-domination number of some classes of graphs. Some of our results are extensions of<br />well-known properties of the signed total Roman $k$-domination number $gamma_{stR}^k(G)$,<br />introduced and investigated by Volkmann [9,12].http://comb-opt.azaruniv.ac.ir/article_14112_2bc938aa277cc581fae498b2671afd4e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Distinct edge geodetic decomposition in graphs1851961411410.22049/cco.2020.26638.1126ENJ.JOHNGoverment College of Engineering, Tirunelveli0000-0001-5528-4387D.StalinBharathiyar UniversityJournal Article20190811Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,<br />(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph <br />G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),<br />(1≤i≠j≤n). The maximum cardinality of π is called the distinct edge geodetic decomposition number of G and is denoted by π_dg1 (G), where g_1 (G) is the edge geodetic number of G. Some general properties satisfied by this concept are studied. Connected graphs which are edge geodetic decomposable are characterized. Connected distinct edge geodetic decomposable graphs of order p with π_dg1 (G)= p-2 are characterised.http://comb-opt.azaruniv.ac.ir/article_14114_8a5d0d833f57f98f5cdcf8e147834b4d.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201A characterization relating domination, semitotal domination and total Roman domination in trees1972091411310.22049/cco.2020.26892.1157ENAbelCabrera MartinezUniversitat Rovira i Virgili, Tarragona, Spain0000-0003-2806-4842AlondraMartinez AriasDepartamento de Matemática, Universidad de Oriente, CubaMaikelMenendez CastilloDepartamento de Matemática, Universidad de Oriente, CubaJournal Article20200727A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_{vin V(G)}f(v)$ among all total Roman dominating functions $f$ on $G$.<br />It is known that $gamma_{tR}(G)geq gamma_{t2}(G)+gamma(G)$ for any graph $G$ with neither isolated vertex nor components isomorphic to $K_2$, where $gamma_{t2}(G)$ and $gamma(G)$ represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.http://comb-opt.azaruniv.ac.ir/article_14113_c873714f5be01511ca842f7635e16a45.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Stirling number of the fourth kind and lucky partitions of a finite set2112191411510.22049/cco.2020.26895.1158ENJohanKokIndependent Mathematics Researcher, City of Tshwane0000-0003-0106-1676Joseph VargheseKureetharaDepartment of Mathematics, Christ University0000-0001-5030-3948Journal Article20200729The concept of Lucky <em>k</em>-polynomials and in particular Lucky <em>χ</em>-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky <em>k</em>- or Lucky <em>χ</em>-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.http://comb-opt.azaruniv.ac.ir/article_14115_a34ab568089a25e5dc708cb1dd13da5e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Total domination in cubic Knodel graphs2212301413310.22049/cco.2020.26793.1143ENNaderJafari RadDoost AliMojdehDeparttment of Mathematics, University of Mazandaran0000-0001-9373-3390RezaMusawiShahrood University of TechnologyE.NazariTafresh UniversityJournal Article20200320A subset <em>D</em> of vertices of a graph <em>G</em> is a <em>dominating set</em> if for each <em>u ∈ V (G) D</em>, <em>u</em> is adjacent to some<br />vertex <em>v ∈ D</em>. The <em>domination number</em>, <em>γ(G)</em> of<br /><em>G</em>, is the minimum cardinality of a dominating set of <em>G</em>. A set<br /><em>D ⊆ V (G)</em> is a <em>total dominating set</em> if for each<br /><em>u ∈ V (G)</em>, <em>u</em> is adjacent to some vertex <em>v ∈ D</em>. The<br /><em>total domination number</em>, <em>γ<sub>t </sub>(G)</em> of <em>G</em>, is the<br />minimum cardinality of a total dominating set of <em>G</em>. For an even<br />integer $nge 2$ and $1le Delta le lfloorlog_2nrfloor$, a<br />Knodel graph $W_{Delta,n}$ is a $Delta$-regular<br />bipartite graph of even order <em>n</em>, with vertices <em>(i,j)</em>, for<br />$i=1,2$ and $0le jle n/2-1$, where for every $j$, $0le jle<br />n/2-1$, there is an edge between vertex $(1, j)$ and every vertex<br />$(2,(j+2^k-1)$ mod (n/2)), for $k=0,1,cdots,Delta-1$. In this<br />paper, we determine the total domination number in $3$-regular<br />Knodel graphs $W_{3,n}$.http://comb-opt.azaruniv.ac.ir/article_14133_4811ad0eba6be31d291108004a13ceab.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201The annihilator-inclusion Ideal graph of a commutative ring2312481413410.22049/cco.2020.26752.1139ENJ.AmjadiAzarbaijan Shahid Madani University0000-0001-9340-4773R.KhoeilarAzarbaijan Shahid Madani University0000-0002-2981-3625A.AlilouJabir Ibn Hayyan research centerJournal Article20200125Let <em>R</em> be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of <em>R</em> , denoted by ξ<sub>R</sub>, is a graph whose vertex set is the of all<br />non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent<br />if and only if either <em>Ann(I) ⊆ J</em> or <em>Ann(J) ⊆ I</em>. In this paper, we investigate the basic<br />properties of the graph ξ<sub>R</sub>. In particular, we show<br />that ξ<sub>R</sub> is a connected graph with diameter at most three, and<br />has girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.http://comb-opt.azaruniv.ac.ir/article_14134_d827d160a865bae7e60ce70ded206a7f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201On the variable sum exdeg index and cut edges of graphs2492571414210.22049/cco.2021.26865.1152ENAnsaKanwalKnowledge Unit of Science, University of Management and Technology,
Sialkot, PakistanAdnanAslamDepartment of Natural Sciences and Humanities,
University of Engineering and Technology, Lahore (RCET), PakistanZahidRazaDepartment of Mathematics, College of Sciences,
University of Sharjah, Sharjah, UAENaveedIqbalDepartment of Mathematics, Faculty of Science, University of Ha'il,
Ha'il, Saudi ArabiaBawfehKometaDepartment of Mathematics, Faculty of Science, University of Ha'il,
Ha'il, Saudi ArabiaJournal Article20200704The variable sum exdeg index of a graph <em>G</em> is defined as $SEI_a(G)=sum_{uin V(G)}d_G(u)a^{d_G(u)}$, where $aneq 1$ is a positive real number, <em>d<sub>u</sub>(u)</em> is the degree of a vertex <em>u ∈ V (G)</em>. In this paper, we characterize the graphs with the extremum variable sum exdeg index among all the graphs having a fixed number of vertices and cut edges, for every <em>a>1</em>.http://comb-opt.azaruniv.ac.ir/article_14142_2b1c0a9533954e332f695518dd3b6b6d.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs2592711414710.22049/cco.2021.26987.1173ENIgorMilovanovicFaculty of Electronic Engineering, Nis, SerbiaEminaMilovanovicFaculty of Electronic EngineeringMarjanMatejicFaculty of Electronic EngineeringSerife BurcuBozkurt AltındağYenikent Kardelen Konutlari, SelcukluJournal Article20201028Let <em>G=(V,E)</em>, $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%<br />n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geq<br />d_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%<br />D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonal<br />degree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}<br />(D+A) {D}^{-1/2}$ the normalized signless Laplacian matrix of graph $G$. The<br />eigenvalues of matrix $mathcal{L}^{+}(G)$, $2=gamma _{1}^{+}geq gamma<br />_{2}^{+}geq cdots geq gamma _{n}^{+}geq 0$, are normalized signless<br />Laplacian eigenvalues of $G$. In this paper some bounds for the sum $%<br />K^{+}(G)=sum_{i=1}^nfrac{1}{gamma _{i}^{+}}$ are considered.http://comb-opt.azaruniv.ac.ir/article_14147_7f8ffc58c035b57d40e721cebe8c5f56.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Outer independent Roman domination number of trees2732861416210.22049/cco.2021.27072.1191ENNasrinDehgardiSirjan University of Technology, Sirjan 78137, IranMChellaliLAMDA-RO Laboratory, Department of Mathematics
University of Blida
B.P. 270, Blida, AlgeriaJournal Article20201002A Roman dominating function (RDF) on a graph <em>G=(V,E)</em> is a function <br /> <em>f : V → {0, 1, 2} </em> such that every vertex <em>u</em> for which <em>f(u)=0</em> is<br /> adjacent to at least one vertex <em>v</em> for which <em>f(v)=2</em>. An RDF <em>f</em> is called<br />an outer independent Roman dominating function (OIRDF) if the set of<br />vertices assigned a 0 under <em>f</em> is an independent set. The weight of an<br />OIRDF is the sum of its function values over all vertices, and the outer<br />independent Roman domination number <em>ΥoiR (G)</em> is the minimum weight<br />of an OIRDF on $G$. In this paper, we show that if <em>T</em> is a tree of order <em>n ≥ 3</em><br /> with <em>s(T)</em> support vertices, then $gamma _{oiR}(T)leq min {<br />frac{5n}{6},frac{3n+s(T)}{4}}.$ Moreover, we characterize the tress<br />attaining each bound.http://comb-opt.azaruniv.ac.ir/article_14162_bdd30b3dcafac6dbd48a2ae823774238.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Strength of strongest dominating sets in fuzzy graphs2872971416310.22049/cco.2021.26988.1174ENNaderJafari RadShahed UniversityMarziehFarhadi JalalvandShahrood University of TechnologyMaryamGhoraniFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, iranJournal Article20201030A set <em>S</em> of vertices in a graph <em>G=(V,E)</em> is a dominating set of<br /><em>G</em> if every vertex of <em>V-S</em> is adjacent to some vertex of <em>S</em>.<br />For an integer <em>k≥1</em>, a set <em>S</em> of vertices is a <em>k</em>-step <br />dominating set if any vertex of $G$ is at distance <em>k</em> from some<br />vertex of <em>S</em>. In this paper, using membership <br />values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongest<br />dominating set as well as strength of strongest<br />$k$-step dominating set in fuzzy graphs. We determine various bounds for<br />these parameters in fuzzy graphs. We also determine the strength<br />of strongest dominating set in some families of fuzzy graphs<br />including complete fuzzy graphs and complete bipartite fuzzy<br />graphs.http://comb-opt.azaruniv.ac.ir/article_14163_74397a887a02a4b6959e0c11b1873018.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Line completion number of grid graph Pn × Pm2993131416510.22049/cco.2021.26884.1156ENJoseph VargheseKureetharaChrist University0000-0001-5030-3948MerinSebastianChrist UniversityJournal Article20200719The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least <em>r</em> edges, the super line graph of index <em>r</em>, <em>Lr(G)</em>, has as its vertices the sets of <em>r</em>-edges of <em>G</em>, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number <em>lc(G)</em> of a graph <em>G</em> is the least positive integer <em>r</em> for which <em>Lr(G)</em> is a complete graph. In this paper, we find the line completion number of grid graph <em>Pn × Pm</em> for various cases of <em>n</em> and <em>m</em>.http://comb-opt.azaruniv.ac.ir/article_14165_1171c3fa79e8d1bdc282acdfcee82cd8.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles3153241416610.22049/cco.2021.27067.1188ENNasrinDehgardiSirjan University of Technology, Sirjan 78137, IranJournal Article20201228Let <em>G</em> be a graph. A 2-rainbow dominating function (or 2-RDF) of <em>G</em> is a function <em>f</em> from <em>V(G)</em> <br />to the set of all subsets of the set <em>{1,2}</em> <br />such that for a vertex <em>v ∈ V (G)</em> with <em>f(v) = ∅</em>, the<br />condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled, wher N<sub>G</sub>(v) is the open neighborhood<br />of <em>v</em>. The weight of 2-RDF <em>f</em> of <em>G</em> is the value<br />$omega (f):=sum _{vin V(G)}|f(v)|$. The 2-rainbow<br />domination number of <em>G</em>, denoted by Υr2 (G), is the<br />minimum weight of a 2-RDF of <em>G</em>. A 2-RDF <em>f</em> is called an outer independent 2-rainbow dominating function (or OI2-RDF) of <em>G</em> if<br />the set of all <em>v ∈ V (G)</em> with <em>f(v) = ∅</em> is an <br />independent set. <br />The outer independent 2-rainbow domination number <em>Υ<sub>oir2 </sub>(G)</em> is<br />the minimum weight of an OI2-RDF of <em>G</em>. <br />In this paper, we obtain the<br />outer independent 2-rainbow domination number of <em>Pm□Pn</em><sub></sub> and <em>Pm□Cn</em>. Also we determine the value of <em>Υ<sub>oir2 </sub>(Cm2Cn)</em> when <em>m</em> or <em>n</em> is even.http://comb-opt.azaruniv.ac.ir/article_14166_29b8f21c8804bd9e727a49997e1ae57c.pdf