Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Independent domination, order, size, and maximum degree107310801503710.22049/cco.2025.30756.2609ENSimoneDantasInstituto de Matemática e EstatÍstica, Universidade Federal Fluminense, Niterói, BrazilMichael A.HenningDepartment of Mathematics and Applied Mathematics, University of Johannesburg,
Auckland Park, 2006, South AfricaGiovannaArelis Baldeon PenaoInstituto de Matemática e EstatÍstica, Universidade Federal Fluminense, Niterói, BrazilDieterRautenbachInstitute of Optimization and Operations Research, Ulm University, Ulm, GermanyJournal Article20250630For a connected graph $G$ with $n>2\Delta$ vertices, $m$ edges, and maximum degree at most $\Delta\geq 3$, we show $i(G)\leq \left(1-\Omega\left(\frac{1}{\Delta^4}\right)\right)n-\frac{m}{\Delta}+O\left(\frac{1}{\Delta^2}\right)$ and discuss related problems. https://comb-opt.azaruniv.ac.ir/article_15037_5228fbb71a2f533756a0da20b9e85b56.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Weak signed total Italian domination in digraphs108110911502810.22049/cco.2025.30748.2606ENLutzVolkmannRWTH Aachen University, , 52056 Aachen, GermanyJournal Article20250628A weak signed total Italian dominating function (WSTIDF) of a digraph $D$ with vertex set $V(D)$ is defined as a<br />function $f:V(D)\rightarrow\{-1,1,2\}$ having the property that $\sum_{x\in N^-(v)}f(x)\ge 1$ for each $v\in V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which arcs go into $v$. The weight of a WSTIDF is the sum of its function values over all vertices. The weak signed total Italian domination number of $D$, denoted by $\gamma_{wstI}(D)$, is the minimum weight of a WSTIDF on $D$. We initiate the study of the weak signed total Italian domination number in digraphs, and we present different sharp bounds on $\gamma_{wstI}(D)$. In addition, we determine the weak signed total Italian domination number of some classes of digraphs.https://comb-opt.azaruniv.ac.ir/article_15028_4163a82155762ca8713bd3d931ee079e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Total domination versus triad domination109311061504010.22049/cco.2025.30952.2676ENTeresa W.HaynesDepartment of Mathematics and Statistics, East Tennessee State University,
Johnson City, TN 37614-0002 USADepartment of Mathematics and Applied Mathematics, University of Johannesburg,
Auckland Park, 2006 South AfricaMichael A.HenningDepartment of Mathematics and Applied Mathematics, University of Johannesburg,
Auckland Park, 2006 South AfricaJournal Article20250826A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A total dominating set in $G$ is a dominating set $S$ with the additional property that the subgraph $G[S]$ induced by $S$ is isolate-free. A triad dominating set $S$ (also called a $3$-component dominating set in the literature) is a dominating set in which every component in $G[S]$ has order at least~$3$. The triad domination number, denoted $\gamma_{td}(G)$, of $G$ is the minimum cardinality among all triad dominating sets of $G$. We observe that $\gamma(G) \le \gamma_t(G) \le \gamma_{td}(G)$, where $\gamma(G)$ is the domination number of $G$ and $\gamma_t(G)$ is the total domination number of $G$. We show that the ratio $\frac{\gamma_{td}(G)}{\gamma_t(G)}$ is at most $\frac{3}{2}$. We establish properties of the graphs $G$ satisfying $\gamma_{td}(G) = \frac{3}{2}\gamma_t(G)$ and characterize the trees achieving this equality.https://comb-opt.azaruniv.ac.ir/article_15040_a0df9bb4e5d9f03fdaa958142cac4205.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Majority Sets in Graphs110711231506210.22049/cco.2025.30739.2602ENMustaphaChellaliAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, AlgeriaStephen T.HedetniemiProfessor and Chair Emeritus of Computer Science, Clemson University, Clemson, SC 29634 USANacéraMeddahAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, AlgeriaJournal Article20250626A set of vertices $S\subseteq V$ in a graph $G=(V,E)$ is called an internal majority set if for every vertex $v\in S$, a majority of the neighbors of $v$ are in $S$, or equivalently, every vertex $v\in S$ has fewer neighbors in $V-S=\overline{S}$ than it has in $S$. A set $S$ is called an external majority set if for every vertex $v\in\overline{S}$, a majority of the neighbors of $v$ are in $S$, or equivalently, every vertex $v\in\overline{S}$ has more neighbors in $S$ than it has in $\overline{S}$. A set of vertices $S\subseteq V$ in a graph $G=(V,E)$ is called a total majority set if for every vertex $v\in V$, a majority of the neighbors of $v$ are in $S$, or equivalently, every vertex $v\in V$ has more neighbors in $S$ than it has in $\overline{S}$. In this paper we show that majority sets in graphs are closely related to, but different than, a variety of sets that have been studied, such as offensive alliances, cost effective and very cost effective sets and unfriendly partitions in graphs. We also prove that the decision problems associated with external majority sets and total majority sets are NP-complete. Finally, we present a list of open problems related to majority sets in graphs.https://comb-opt.azaruniv.ac.ir/article_15062_7a76026083da7727f55d23a2eca85043.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201On 4-domination and 4-rainbow domination of cylindrical graphs112511401506310.22049/cco.2025.30780.2616ENJanezŽerovnikFME, University of Ljubljana, Aškerčeva 6, Ljubljana, 1000, SloveniaRudolfovo - Science and Technology Centre Novo Mesto,
Podbreznik 15, Novo mesto, 8000, SloveniaJournal Article20250709Cylindrical graphs and torus grid graphs are naturally constructed from sub-graphs of the infinite grid by certain identifications of boundary vertices. Considering various domination type problems, it is usually possible to find an optimal solution on the infinite grid. To the contrary, exact values of invariants for the cylindrical and torus grid graphs are typically only known for special subfamilies, and are in general hard to compute. The 4-domination and 4-rainbow domination of cylindrical graphs is studied, and some new formulae and improved bounds are reported, generalizing recent results for the case $k = 2$ in [Computational and Applied Mathematics 44(5), 293 (2025)]. We also consider weak 4-domination and singleton 4-rainbow domination.https://comb-opt.azaruniv.ac.ir/article_15063_298f446d63ac403ca9ba7b34cf485d68.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201On the global Italian domination of graphs114111551507510.22049/cco.2025.30937.2670ENGuoliangHaoSchool of Mathematics and Statistics, Heze University, Heze 274015, Shandong, ChinaZhihongXieSchool of Business, Heze University, Heze 274015, Shandong, ChinaYuqiWuSchool of Computer Information Engineering, Nanchang Institute of Technology, Nanchang 330044, Jiangxi, P.R. ChinaSeyed MahmoudSheikholeslamiDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. IranJournal Article20250824Let $H$ be a graph with vertex set $V.$ An Italian dominating function (IDF) on $H$ is a function from $V$ to the set $\{0,1,2\}$ having the property that any vertex assigned $0$ is adjacent to two vertices assigned $1$ or one vertex assigned $2.$ The value $\sum_{x\in V}h(x)$ is called the weight of an IDF $h$ on $H.$ A global Italian dominating function (GIDF) on $H$ is an IDF on $H$ and its complement. The minimum weight of an IDF (resp., GIDF) on $H$ is the Italian (resp., global Italian) domination number of $H.$ In this paper, we establish several relations between the global Italian domination and Italian domination numbers. In particular, we determine the difference between these two parameters of cubic graphs.https://comb-opt.azaruniv.ac.ir/article_15075_0c701829ef369cf9cc1ed36488cfa4cd.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201A Note on Distance-Fall Colorings115711611508810.22049/cco.2026.30962.2683ENWayneGoddardSchool of Mathematical and Statistical Sciences Clemson University, USASonwabileMafundaSoka University of America, USAUniversity of Johannesburg, South AfricaJournal Article20250829We say a proper coloring of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, then it has a distance-2 fall 3-coloring. Further, for every integer $k\ge 2$, if $T$ is a tree of order at least $k$, then $T$ has a $k$-coloring such that every vertex is within distance $k-1$ of every color. This proves an old conjecture of Beineke and Henning that every tree of order $n$ has an independent distance-$d$-dominating set of size at most $n/(d + 1)$.https://comb-opt.azaruniv.ac.ir/article_15088_335ba1e7ff9537a3161ff540c7c78ce7.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Upper bounds for $[1,2]$-domination number in trees116311741512810.22049/cco.2026.31357.2831ENJ.AmjadiDepartment of Mathematics,
Azarbaijan Shahid Madani University,
Tabriz, I.R. IranM.EbadiDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. IranSeyed MahmoudSheikholeslamiDepartment of Mathematics,
Azarbaijan Shahid Madani University,
Tabriz, I.R. IranL.VolkmannInstitute for Geometry and Practical Mathematics, RWTH Aachen University,
52056 Aachen, GermanyJournal Article20251223A set $S$ of vertices is a $[1,2]$-set of a graph $G$ if every vertex $v$ not in $S$ is adjacent to at least one but no more than two vertices in $S$. The minimum cardinality of a $[1,2]$-set is the $[1,2]$-domination number. In this paper, we present two upper bounds on the $[1,2]$-domination number of trees in terms of the order, number of support vertices and number of leaves. Furthermore, extremal trees reaching one of these two bounds are provided.https://comb-opt.azaruniv.ac.ir/article_15128_2250a82aa247078a96542171dcafe737.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201A note on independent domination in almost-regular graphs117511801513210.22049/cco.2026.31146.2756ENWayneGoddardSchool of Mathematical and Statistical Sciences, Clemson University, Clemson, USADepartment of Mathematics and Applied Mathematics, University of Johannesburg Auckland Park, 2006 South AfricaMichael A.HenningDepartment of Mathematics and Applied Mathematics, University of Johannesburg Auckland Park, 2006 South AfricaJournal Article20251024A classic result in domination theory is that a regular graph has independent domination number at most half the order. We strengthen this result to ``almost-regular'' graphs by showing that if a graph has minimum degree $\delta > 0$ and maximum degree at most $\delta + 3$, and the subgraph induced by the vertices of degree $\delta + 3$ (if any) is bipartite, then the independent domination number is at most half the order. We also discuss related questions.https://comb-opt.azaruniv.ac.ir/article_15132_c76ef09a70b3aba61b01d8a53b8439c4.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Breaking Symmetry in Graphs by Resolving Sets118111951513310.22049/cco.2026.30820.2632ENNasrinSoltankhahDepartment of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, IranMeysamKorivandDepartment of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, IranSandiKlavžarFaculty of Mathematics and Physics, University of Ljubljana, SloveniaJournal Article20250722Let $dim(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le dim(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars attain the bound, and among connected unicyclic graphs such graphs are $t$-cycles for $t\in \{3,4,5\}$. It is shown that for any $1\leq n< m$, there exists a graph $G$ with $D(G)=n$ and ${\rm dim}(G)=m$. Using the bound $D(G) \le dim(G)+1$, graphs with $D(G) = n(G)-2$ are classified. https://comb-opt.azaruniv.ac.ir/article_15133_865c85d6dc7a3f8f56b24284c6611b43.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Independent location-domination number of graphs119712131513710.22049/cco.2026.30238.2378ENPailinKaewpermDepartment of Mathematics, Faculty of Science,
King Mongkut’s University of Technology Thonburi, Bangkok, ThailandMathematics and Statistics with Applications (MaSA), Bangkok, ThailandDavid A.KalarkopDepartment of Mathematics, St Joseph’s University, Bengaluru, IndiaPawatonKaemawichanuratDepartment of Mathematics, Faculty of Science,
King Mongkut’s University of Technology Thonburi, Bangkok, ThailandMathematics and Statistics with Applications (MaSA), Bangkok, ThailandJournal Article20250101Let $G = (V(G), E(G))$ be a graph. A set $I \subseteq V(G)$ is independent if no two vertices of $I$ are adjacent. A set $D \subseteq V(G)$ is dominating if every vertex $u \in V(G) \setminus D$ is adjacent to a vertex in $D$. A set $L \subseteq V(G)$ is an independent locating-dominating set (ILD-set) of $G$ if $L$ is independent and dominating with the additional property that $N (u) \cap L \neq N (v) \cap L$ for any pair of distinct $u, v \in V(G) \setminus L$. The independent location-domination number of a graph $G$ is the minimum cardinality of an ILD-set of $G$ and is denoted by $i_{\ell}(G)$. In this paper, we study the non-existence of ILD-sets of maximal outerplanar graphs and circulants graphs. In trees, we prove that \textcolor{red}{$\frac{n + 1}{3} \leq i_{\ell}(T) \leq n - 1$} for every tree $T$ of $n$ vertices. We further prove that there exists a tree $T$ with prescribed value $i_{\ell}(T)$ between these bounds.https://comb-opt.azaruniv.ac.ir/article_15137_62b8fc5cd71a689649cb3518c929fee2.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201A note on the domination number in bipartite graphs121512191514710.22049/cco.2026.31447.2855ENNaderJafari RadDepartment of Mathematics, Shahed University, Tehran, IranJournal Article20260101Archdeacon et al. [J. Graph Theory 46 (2004), 207--210] proved that if $G$ is a bipartite graph with partite sets $X$ and $Y$ whose vertices in $Y$ are of minimum degree at least $3$ then there exists a set $A\subseteq X$ of size at most<br />$\frac{|X\cup Y|}{4}$ such that every vertex in $Y$ is adjacent to a vertex in $A$. We generalize this result for all bipartite graphs with minimum degree $\delta\geq 3$ using the Brooks Theorem on the vertex coloring.https://comb-opt.azaruniv.ac.ir/article_15147_9f617b1b10337f51265d1ac407e11d02.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Bounding the Eviction Number of a Graph in Terms of its Independence Number122112341516610.22049/cco.2026.31031.2719ENGaryMacGillivrayDepartment of Mathematics and Statistics, University of Victoria, Victoria, CanadaChristina M.MynhardtDepartment of Mathematics and Statistics, University of Victoria, Victoria, CanadaVirgelotVirgileDepartment of Mathematics and Statistics, University of Victoria, Victoria, CanadaJournal Article20250924An eternal dominating family of a graph $G$ in the eviction game is a collection $\mathcal{D}_{k}=\{D_{1},D_{2},\dots,D_{l}\}$ of dominating sets of $G$ such that (a) $|D_{i}|=|D_{j}|$ for all $i,j\in\{1,2,\dots,l\}$, and (b) for any $i\in \{1,2,\dots,l\}$ and any $v\in D_{i}$, either all neighbours of $v$ belong to $D_{i}$, or there are a neighbour $w$ of $v$ not in $D_{i}$ and an integer $j\in\{1,2,\dots,l\}\setminus\{i\}$ such that $D_{i}\cup\{w\}\setminus \{v\}=D_{j}$. The eviction number of $G$, denoted by $e^{\infty}(G)$, is the smallest cardinality of the sets in such an eternal dominating family.<br />We compare $e^{\infty}$ to the independence number $\alpha$. We show that the ratio $\alpha/e^{\infty}$ is unbounded and construct an infinite class of connected graphs for which $e^{\infty}/\alpha \approx 4/3$. As our main result, we use Ramsey numbers to show that for any integer $k\geq1$, there exists a function $f(k)$ such that any graph with independence number$k$ has eviction number at most $f(k)$.https://comb-opt.azaruniv.ac.ir/article_15166_f0552c5dd4d26deaa32fdb880b67d435.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201New Characterization of Efficient Closed and Open Dominated Graphs123512471517610.22049/cco.2026.31349.2826ENVeronicaHernandez MartinezUniversidad Carlos III de Madrid, Madrid, SpainIztokPeterinFaculty of Electrical Engineering and Computer Science, University of Maribor, SloveniaInstitute of Mathematics, Physics and Mechanics, Ljubljana, SloveniaJournal Article20251220A graph $G$ is an efficient closed dominated graph (ECD-graph) if there exists a subset of vertices whose closed neighborhoods partition $V(G)$ and is an efficient open dominated graph (EOD-graph) if there exists a subset of vertices whose open neighborhoods partition $V(G)$. We present a new characterization of ECD- and EOD-graphs that involves independent number and a vertex clique cover of some family of cliques of closed neighborhood graph and open neighborhood graph, respectively, that are intersection graphs of closed and open neighborhoods, respectively. Several consequences are presented as well, one of them with respect to the Vizing's conjecture and the other solves a conjecture on EOD-graphs among toruses $C_t\Box C_r$ posed by Kuziak et al. (Discrete Math. Theoret. Comput. Sci. 16 (2014) 105-120).https://comb-opt.azaruniv.ac.ir/article_15176_1e8c4feddd383909a6c419b363c14461.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201Maker-Breaker total domination number124912631517710.22049/cco.2026.30824.2634ENAthiraDivakaranDepartment of Mathematics, Mar Athanasius College, Kothamangalam, IndiaTijoJamesDepartment of Mathematics, Pavanatma College, Murickassery, IndiaSandiKlavžarFaculty of Mathematics and Physics, University of Ljubljana, SloveniaInstitute of Mathematics, Physics and Mechanics, Ljubljana, SloveniaFaculty of Natural Sciences and Mathematics, University of Maribor, SloveniaLatha SNairDepartment of Mathematics, Mar Athanasius College, Kothamangalam, IndiaJournal Article20250723The Maker-Breaker total domination number, $\gamma_{\rm MBT}(G)$, of a graph $G$ is introduced as the minimum number of moves of Dominator to win the Maker-Breaker total domination game, provided that he has a winning strategy and is the first to play. The Staller-start Maker-Breaker total domination number, $\gamma_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $\gamma_{\rm MBT}(G)$ and on $\gamma_{\rm MBT}'(G)$ are provided and demonstrated to be sharp. It is proved that for any pair of integers $(k,\ell)$ with $2\leq k\leq \ell$, (i) there exists a connected graph $G$ with $\gamma_{\rm MB}(G)=k$ and $\gamma_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $\gamma_{\rm MB}'(G')=k$ and $\gamma_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $\gamma_{\rm MBT}(G'')=k$ and $\gamma_{\rm MBT}'(G'')=\ell$. Here, $\gamma_{\rm MB}$ and $\gamma_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game.https://comb-opt.azaruniv.ac.ir/article_15177_0972e39f2250a6f43cb2b34e6e32d563.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201On maximizing private neighbors in graphs126512791518110.22049/cco.2026.31182.2771ENStephen T.HedetniemiEmeritus Professor of Computer Science, Clemson University, Clemson, SC, USADouglasRallEmeritus Professor of Mathematics, Furman University, Greenville, SC, USAJournal Article20251103Given a set $U \subset V$ of vertices in a graph $G = (V, E)$, a {\it private neighbor with respect to the set $U$} is any vertex $w \in V$ having precisely one neighbor, say $v$, in $U$. If $w \in V - U$, then $w$ is called an {\it external private neighbor} of $v$ with respect to $U$. If $w \in U$ then $w$ is called an {\it internal private neighbor} of $v$ with respect to $U$. We also add one special case: if $w \in U$ and $N(w) \cap U = \emptyset$, then we say that $w$ is a {\it self private neighbor} with respect to $U$. By definition, a self private neighbor with respect to $U$ is an isolated vertex in the subgraph of $G$ induced by $U$. In this paper we consider the general problems of trying to find sets of vertices which maximize the number of private neighbors of specific types in a graph. In the process of doing this we define several new maximization parameters of graphs which generalize some known and well-studied parameters of graphs relating to vertex and edge independence, domination and irredundance in graphs.https://comb-opt.azaruniv.ac.ir/article_15181_ecb50e26f7f3be72f147f9eb97b99bc4.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201On proximity and other distance parameters in planar graphs128113041518410.22049/cco.2026.30964.2684ENPeterDankelmannUniversity of Johannesburg, South AfricaSonwabileMafundaSoka University in America, USASufiyanMalluUniversity of Johannesburg, South AfricaJournal Article20250829Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum, respectively, of the average distances of the vertices of $G$.<br />It was shown by Aouchiche and Hansen [Proximity and remoteness in graphs: bounds and conjectures, Networks 58 no. 2 (2011)] that for a connected graph of order $n$, the difference between remoteness and proximity and the difference between radius and proximity are bounded from above by about $\frac{n}{4}$, and the difference between diameter and proximity is bounded from above by about $\frac{3}{4}n$.<br />In this paper, we show that all three bounds can be improved significantly for simple triangulations (i.e., triangulations), and for graphs of given connectivity.<br />We show that in simple triangulations the above bound on the difference between radius and proximity can be improved to about $\frac{1}{12}n$, and further to about $\frac{1}{16}n$ and $\frac{1}{20}n$ if the graphs is, in addition, $4$-connected or $5$-connected, respectively. Similar improvements are shown for simple quadrangulations (i.e., maximal bipartite planar graphs), and for maximal outerplanar graphs. We further show that the above bound on the difference between remoteness and proximity can be improved to about $\frac{1}{4\kappa}n$ if $G$ is $\kappa$-connected.<br />Finally, we improve the bound on the difference between diameter and proximity to about $\frac{3}{4\kappa}n$ if $G$ is $\kappa$-connected. We present graphs that demonstrate that our bounds are either sharp, or sharp apart from an additive constant, even if restricted to planar graphs.https://comb-opt.azaruniv.ac.ir/article_15184_cbac92d4e3f0ca6dc4fc136f58136311.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201The hamiltonicity and pancyclicity of split graphs130513171518810.22049/cco.2026.31342.2822ENJunqingCaiSchool of Mathematical Science, Tianjin Normal University, Tianjin 300387, ChinaInstitute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University,
Tianjin 300387, ChinaHaoLiLaboratoire Interdisciplinaire des Sciences du Num´erique,
UMR9015 CNRS and Universit´e Paris-Saclay, Campus Universitaire, Orsay 91405, FranceZhiyiJiangSchool of Mathematical Science, Tianjin Normal University, Tianjin 300387, ChinaJournal Article20251219A split graph is a graph whose vertex set can be partitioned into two disjoint subsets (either of which may be empty) such that one subset induces a clique and the other induces an independent set. Regarding the hamiltonicity of such graphs, Dai et al. [Discrete Math. 345 (2022), 112826] conjectured that every $r$-connected $K_{1, r+1}$-free split graph is hamiltonian. In this paper, we provide a partial verification of this conjecture for the case $r=4$. Precisely, we show that every $4$-connected $\{K_{1,5}, K_{1,5}+e\}$-free split graph is hamiltonian.<br /> <br />Furthermore, we address Bondy’s meta-conjecture proposed in 1971, which asserts that almost any nontrivial condition guaranteeing a graph to be hamiltonian also implies the graph to be pancyclic, except for a small number of well-characterized exceptional graphs. We prove that this meta-conjecture holds for split graphs.https://comb-opt.azaruniv.ac.ir/article_15188_ac00fb70d9bbafe852f824cd1e26de99.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201On graphs having proper $(1; k)$-dominating sets131913311519410.22049/cco.2026.31244.2794ENMostafaBlidiaLAMDA-RO Laboratory, Department of Mathematics, University of Blida,
B.P. 270, Blida, AlgeriaAhmedBouchouLAMDA-RO Laboratory, Department of Mathematics, University of Blida,
B.P. 270, Blida, AlgeriaUniversity of Médéa, AlgeriaJournal Article20251121A (1,k)-dominating set, denoted (1,k)-dset, in a graph G=(V,E) is a set S having the property that for every vertex v in V-S, there is at least one vertex in S within distance 1 from v and a second vertex in S within distance at most k from v. A proper (1,k)-dominating set, denoted (1,k)-dset, in a graph G=(V,E) is a set D that is (1,k)-dset but not (1,k-1)-dset, meaning that D is a (1,k)-dset and there is at least one vertex v in V-D that has exactly one vertex in D within distance 1 from v, no vertices in D within distance k-1 from v and there exists at least one other vertex in D within distance k from v. The (1,k)-domination number (the proper (1,k)-domination number, respectively) of a graph G, denoted gamma_{1,k}(G) (gamma_{1,k bar}(G), respectively) is the minimum cardinality of a (1,k)-dset (a (1,k)-dset, respectively) in G. In this paper, we are interested in the study and existence of (1,k)-dsets in graphs We start by giving a characterization of graphs having (1,k)-dsets for k in {3,4}. Next, we study triangle-free graphs G with gamma_{1,k}(G)=gamma_{1,k bar}(G) for k in{3,4}. Finally, we study the complexity of the (1,k)-domination number and the (1,k bar)-domination number in bipartite graphs.https://comb-opt.azaruniv.ac.ir/article_15194_0a8f395097cc4ce96161e9d9e461024a.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212811420261201The Strong Total Roman Domination in Fuzzy Graphs133313571519810.22049/cco.2026.31398.2841ENMartinCeraDepartamento de Matemática Aplicada I, Universidad de Sevilla, SpainPedroGarcia-VazquezDepartamento de Matemática Aplicada I, Universidad de Sevilla, SpainJuan CarlosValenzuela TripodoroDepartamento de Matemáticas, Universidad de Cádiz, Algeciras Campus, SpainJournal Article20260104In recent years, domination theory and its variants, including Roman domination, have been widely studied in fuzzy graphs due to their ability to model uncertainty in complex networks such as social, transportation, and biological systems. A strong total Roman dominating function (STRDF) on a fuzzy graph $G=(V,\sigma,\mu)$ is a mapping $f:V\rightarrow \{0,1,2\}$ such that every vertex $u$ with $f(u)=0$ has a strong neighbor labeled 2, and every vertex labeled 1 or 2 has at least one strong neighbor with a non-zero label. The strong total Roman domination number, $\gamma_{sR}^t(G)$, is defined as the minimum weight $\sum_{u\in V} f(u)\mu_s(u)$ among all STRDFs $f,$ where $\mu_s(u)$ denotes the minimum membership value of the strong edges incident to $u$. In this paper, we introduce and study the strong total Roman domination number for fuzzy graphs. We establish several bounds, investigate its realizability, determine exact values for several standard families of fuzzy graphs, and present some applications.https://comb-opt.azaruniv.ac.ir/article_15198_fcbc13ac559ebddfd2158464a97d2eab.pdf