Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Signed total Italian k-domination in graphs 171 183 14112 10.22049/cco.2020.26919.1164 EN Lutz Volkmann RWTH Aachen University Journal Article 2020 08 19 Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. A signed total Italian $k$-ominating function (STIkDF) on a graph $G$ is a function $f:V(G)\rightarrow\{-1,1,2\}$ satisfying the conditions that $\sum_{x\in N(v)}f(x)\ge k$ for each vertex $v\in V(G)$, where $N(v)$ is the neighborhood of $v$, and each vertex $u$ with $f(u)=-1$ is adjacent to a vertex $v$ with $f(v)=2$ or to two vertices $w$ and $z$ with $f(w)=f(z)=1$. The weight of an STIkDF $f$ is $\omega(f)=\sum_{v\in V(G)}f(v)$. The signed total Italian $k$-domination number $\gamma_{stI}^k(G)$ of $G$ is the minimum weight of an STIkDF on $G$. In this paper we initiate the study of the signed total Italian $k$-domination number of graphs, and we present different bounds on $\gamma_{stI}^k(G)$. In addition, we determine the signed total Italian $k$-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed total Roman $k$-domination number $\gamma_{stR}^k(G)$, introduced and investigated by Volkmann [9,12].

https://comb-opt.azaruniv.ac.ir/article_14112_c58a0b3d2443dd19b119a3177bde3a4c.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Distinct edge geodetic decomposition in graphs 185 196 14114 10.22049/cco.2020.26638.1126 EN J. JOHN Goverment College of Engineering, Tirunelveli D. Stalin Bharathiyar University Journal Article 2019 08 11 Let $G = (V, E)$ be a simple connected graph of order $p$ and size $q$. A decomposition of a graph $G$ is a collection $\pi$ of edge-disjoint subgraphs $G_{1}, G_{2} ,\dots, G_{n}$ of $G$ such that every edge of $G$ belongs to exactly one $G_{i},(1\leq i\leq n)$. The decomposition $\pi=\{G_{1},G_{2},\dots,G_{n}\}$ of a connected graph $G$ is said to be a distinct edge geodetic decomposition if $g_{1}(G_{i})\neq g_{1}(G_{j}),(1\leq i\neq j\leq n)$. The maximum cardinality of $\pi$ is called the distinct edge geodetic decomposition number of $G$ and is denoted by $\pi_{dg_{1}}(G)$, where $g_{1}(G)$ is the edge geodetic number of $G$. Some general properties satisfied by this concept are studied. Connected graphs of $\pi_{dg_{1}}(G)\geq2$ are characterized and connected graphs of order $p$ with $\pi_{dg_{1}}(G)=p-2$ are characterized.

https://comb-opt.azaruniv.ac.ir/article_14114_9555441344465ec52e1ff6583ab21566.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 A characterization relating domination, semitotal domination and total Roman domination in trees 197 209 14113 10.22049/cco.2020.26892.1157 EN Abel Cabrera Martinez Universitat Rovira i Virgili, Tarragona, Spain Alondra Martinez Arias Departamento de Matemática, Universidad de Oriente, Cuba Maikel Menendez Castillo Departamento de Matemática, Universidad de Oriente, Cuba Journal Article 2020 07 27 A total Roman dominating function on a graph $G$ is a function $f: V(G) \rightarrow \{0,1,2\}$ such that for every vertex $v\in V(G)$ with $f(v)=0$ there exists a vertex $u\in V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set $\{x\in V(G): f(x)\geq 1\}$ has no isolated vertices. The total Roman domination number of $G$, denoted $\gamma_{tR}(G)$, is the minimum weight $\omega(f)=\sum_{v\in V(G)}f(v)$ among all total Roman dominating functions $f$ on $G$. 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.

https://comb-opt.azaruniv.ac.ir/article_14113_0b4f6cdb301bc12d4c4eb2b4ad659382.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Stirling number of the fourth kind and lucky partitions of a finite set 211 219 14115 10.22049/cco.2020.26895.1158 EN Johan Kok Independent Mathematics Researcher, City of Tshwane Joseph Varghese Kureethara Department of Mathematics, Christ University Journal Article 2020 07 29 The concept of Lucky k-polynomials and in particular Lucky χ-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 k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

https://comb-opt.azaruniv.ac.ir/article_14115_627b858f0bf943ab54b52dab41740340.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Total domination in cubic Knödel graphs 221 230 14133 10.22049/cco.2020.26793.1143 EN Nader Jafari Rad Shahed University Doost Ali Mojdeh Departtment of Mathematics, University of Mazandaran Reza Musawi Shahrood University of Technology E. Nazari Tafresh University Journal Article 2020 03 20 A subset $D$ of vertices of a graph $G$ is a dominating set if for each $u\in V(G) \setminus D$, $u$ is adjacent to some vertex $v\in D$. The domination number, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. A set $D\subseteq V(G)$ is a total dominating set if for each $u\in V(G)$, $u$ is adjacent to some vertex $v\in D$. The total domination number, $\gamma_{t}(G)$ of $G$, is the minimum cardinality of a total dominating set of $G$. For an even integer $n\ge 2$ and $1\le \Delta \le \lfloor\log_2n\rfloor$, a Kn\"odel graph $W_{\Delta,n}$ is a $\Delta$-regular bipartite graph of even order $n$, with vertices $(i,j)$, for $i=1,2$ and $0\le j\le \frac{n}{2}-1$, where for every $j$, $0\le j\le \frac{n}{2}-1$, there is an edge between vertex $(1, j)$ and every vertex $(2,(j+2^k-1)$ mod $\frac{n}{2}$), for $k=0,1,\dots,\Delta-1$. In this paper, we determine the total domination number in $3$-regular Kn\"odel graphs $W_{3,n}$.

https://comb-opt.azaruniv.ac.ir/article_14133_065f302c60f2be489fc8794cc9dbebc0.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 The annihilator-inclusion Ideal graph of a commutative ring 231 248 14134 10.22049/cco.2020.26752.1139 EN J. Amjadi Azarbaijan Shahid Madani University R. Khoeilar Azarbaijan Shahid Madani University A. Alilou Jabir Ibn Hayyan research center Journal Article 2020 01 25 Let $R$ be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of $R$, denoted by $\xi_R$, is a graph whose vertex set is the of all non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either ${\rm Ann}(I)\subseteq J$ or ${\rm Ann}(J)\subseteq I$. The purpose of this paper is to provide some basic properties of the graph $\xi_R$. In particular, shows that $\xi_R$ is a connected graph with diameter at most three, and has girth 3 or $\infty$. Furthermore, is determined all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

https://comb-opt.azaruniv.ac.ir/article_14134_fabb72cabeefd582e42d9da931cd18fe.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 On the variable sum exdeg index and cut edges of graphs 249 257 14142 10.22049/cco.2021.26865.1152 EN Ansa Kanwal Knowledge Unit of Science, University of Management and Technology, Sialkot, Pakistan Adnan Aslam Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), Pakistan Zahid Raza Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, UAE Naveed Iqbal Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il, Saudi Arabia Bawfeh Kometa Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il, Saudi Arabia Journal Article 2020 07 04 The variable sum exdeg index of a graph $G$ is defined as $SEI_a(G)=\sum_{u\in V(G)}d_G(u)a^{d_G(u)}$, where $a\neq 1$ is a positive real number, $d_G(u)$ is the degree of a vertex $u\in V(G)$. 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 $a>1$.

https://comb-opt.azaruniv.ac.ir/article_14142_22abcc9eb4c4e3d46d7547d2a67702ad.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs 259 271 14147 10.22049/cco.2021.26987.1173 EN Igor Milovanovic Faculty of Electronic Engineering, Nis, Serbia Emina Milovanovic Faculty of Electronic Engineering Marjan Matejic Faculty of Electronic Engineering Serife Burcu Bozkurt Altındağ Yenikent Kardelen Konutlari, Selcuklu Journal Article 2020 10 28 Let $G=(V,E)$, $V=\{v_1,v_2,\ldots,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\geq d_2\geq\cdots\geq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{n\times n}$ and ${D}=\mathrm{diag}d_1,d_2,\ldots , d_n)$ be the adjacency and the diagonal degree matrix of $G$, respectively. Denote by ${\mathcal{L}^+}(G)={D}^{-1/2} (D+A) {D}^{-1/2}$ the normalized signless Laplacian matrix of graph $G$. The eigenvalues of matrix $\mathcal{L}^{+}(G)$, $2=\gamma _{1}^{+}\geq \gamma_{2}^{+}\geq \cdots \geq \gamma_{n}^{+}\geq 0$, are normalized signless Laplacian eigenvalues of $G$. In this paper some bounds for the sum $K^{+}(G)=\sum_{i=1}^n\frac{1}{\gamma _{i}^{+}}$ are considered.

https://comb-opt.azaruniv.ac.ir/article_14147_4e63e49850d86af85c9864e131513b86.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Outer independent Roman domination number of trees 273 286 14162 10.22049/cco.2021.27072.1191 EN Nasrin Dehgardi Sirjan University of Technology, Sirjan 78137, Iran M Chellali LAMDA-RO Laboratory, Department of Mathematics University of Blida B.P. 270, Blida, Algeria Journal Article 2020 10 02 A Roman dominating function (RDF) on a graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,2\}$ 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 $\gamma _{oiR}(G)$ is the minimum weight of an OIRDF on $G$. In this paper, we show that if $T$ is a tree of order $n\geq 3$ with $s(T)$ support vertices, then $\gamma _{oiR}(T)\leq \min \{\frac{5n}{6},\frac{3n+s(T)}{4}\}.$ Moreover, we characterize the tress attaining each bound.

https://comb-opt.azaruniv.ac.ir/article_14162_bc649d348f3b3863bd5517b8d106538d.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Strength of strongest dominating sets in fuzzy graphs 287 297 14163 10.22049/cco.2021.26988.1174 EN Marzieh Farhadi Jalalvand Shahrood University of Technology Nader Jafari Rad Shahed University Maryam Ghorani Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, iran Journal Article 2020 10 30 A set S of vertices in a graph G=(V,E) is a dominating set of G if every vertex of V-S is adjacent to some vertex of S. For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from some vertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongest dominating set as well as strength of strongest $k$-step dominating set in fuzzy graphs. We determine various bounds for these parameters in fuzzy graphs. We also determine the strength of strongest dominating set in some families of fuzzy graphs including complete fuzzy graphs and complete bipartite fuzzy graphs.

https://comb-opt.azaruniv.ac.ir/article_14163_39e9cbd30c7e18a6c6c3017a77610dd4.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 Line completion number of grid graph Pn × Pm 299 313 14165 10.22049/cco.2021.26884.1156 EN Joseph Varghese Kureethara Christ University Merin Sebastian Christ University Journal Article 2020 07 19 The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least $r$ edges, the super line graph of index $r$, $L_r(G)$, has as its vertices the sets of $r$-edges of $G$, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number $lc(G)$ of a graph $G$ is the least positive integer $r$ for which $L_r(G)$ is a complete graph. In this paper, we find the line completion number of grid graph $P_n \times P_m$ for various cases of $n$ and $m$.

https://comb-opt.azaruniv.ac.ir/article_14165_3168ef556251b68f7f57b9cb0c2c7e96.pdf

Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 2 2021 12 01 On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles 315 324 14166 10.22049/cco.2021.27067.1188 EN Nasrin Dehgardi Sirjan University of Technology, Sirjan 78137, Iran Journal Article 2020 12 28 Let $G$ be a graph. A $2$-rainbow dominating function (or {\em 2-RDF}) of $G$ is a function $f$ from $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that for a vertex $v\in V(G)$ with $f(v)=\emptyset$, the condition $\bigcup_{u\in N_{G}(v)}f(u)=\{1,2\}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The weight of 2-RDF $f$ of $G$ is the value $\omega (f):=\sum _{v\in V(G)}|f(v)|$. The {\em $2$-rainbow domination number} of $G$, denoted by $\gamma_{r2}(G)$, 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 $v\in V(G)$ with $f(v)=\emptyset$ is an independent set. The outer independent $2$-rainbow domination number $\gamma_{oir2}(G)$ is the minimum weight of an OI2-RDF of $G$. In this paper, we obtain the outer independent $2$-rainbow domination number of $P_{m}\square P_{n}$ and $P_{m}\square C_{n}$. Also we determine the value of $\gamma_{oir2}(C_{m}\Box C_{n})$ when $m$ or $n$ is even.

https://comb-opt.azaruniv.ac.ir/article_14166_529748fc23cb8af458c78bcc4dcc2d3a.pdf