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Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with \emph{chromatic number} at most~$k$, we prove that if $G$ is $K_{1,r}$-free, then $\alpha_{\mathcal{F}}(G) \le (r-1)k\gamma(G)$, where $\gamma(G)$ is the \emph{domination number}. When $\mathcal{F}$ is the family of empty graphs, this bound simplifies to $\alpha(G) \le 2\gamma(G)$ for $K_{1,3}$-free (claw-free) graphs, where $\alpha(G)$ is the \emph{independence number} of $G$. For $d$-regular graphs, this is further refined to the bound $\alpha(G) \le 2\left(\frac{d+1}{d+2}\right)\gamma(G)$, which is tight for $d \in \{2, 3, 4\}$. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for $K_{1,r}$-free graphs, we have $\alpha_{\mathcal{F}}(G) \le r(K_r, \mathcal{F^*})\gamma(G)$, where $\mathcal{F^*}$ is the set of graphs not in $\mathcal{F}$. Specializing to $K_q$-free graphs, we show $\alpha_{\mathcal{F}}(G) \le (r(K_q, K_r) - 1)\gamma(G)$. Finally, for the \emph{$k$-independence number} $\alpha_k(G)$, we prove that if $G$ is $K_{1,r}$-free with order $n$ and minimum degree $\delta \ge k+1$, \[ \alpha_k(G) \le \left( \frac{(r-1)(k+1)}{\delta - k + (r-1)(k+1)} \right) n, \] and this bound is sharp for all parameters.

Let $p \in \mathbb{N}$ and $q \in \mathbb{N} \cup \lbrace \infty \rbrace$. We study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of blue vertices, with all remaining vertices colored white. If a white vertex~$v$ has at least~$p$ blue neighbors and at least one of these blue neighbors of~$v$ has at most~$q$ white neighbors, then by the spreading color change rule the vertex~$v$ is recolored blue. The initial set $S$ of blue vertices is a $(p,q)$-spreading set for $G$ if by repeatedly applying the spreading color change rule all the vertices of $G$ are eventually colored blue. The $(p,q)$-spreading set is a generalization of the well-studied concepts of $k$-forcing and $r$-percolating sets in graphs. For $q \ge 2$, a $(1,q)$-spreading set is exactly a $q$-forcing set, and the $(1,1)$-spreading set is a $1$-forcing set (also called a zero forcing set), while for $q = \infty$, a $(p,\infty)$-spreading set is exactly a $p$-percolating set. The $(p,q)$-spreading number, $\sigma_{(p,q)}(G)$, of $G$ is the minimum cardinality of a $(p,q)$-spreading set. In this paper, we study $(p,q)$-spreading in claw-free cubic graphs. While the zero-forcing number of claw-free cubic graphs was studied earlier, for each pair of values $p$ and $q$ that are not both $1$ we either determine the $(p,q)$-spreading number of a claw-free cubic graph $G$ or show that $\sigma_{(p,q)}(G)$ attains one of two possible values.

The balance game is played on a graph $G$ by two players, Admirable (A) and Impish (I), who take turns selecting unlabeled vertices of $G$. Admirable labels the selected vertices by $0$ and Impish by $1$, and the resulting label on any edge is the sum modulo $2$ of the labels of the vertices incident to that edge. Let $e_0$ and $e_1$ denote the number of edges labeled by $0$ and $1$ after all the vertices are labeled. The discrepancy in the balance game is defined as $d = e_1 - e_0$. The two players have opposite goals: Admirable attempts to minimize the discrepancy $d$ while Impish attempts to maximize $d$. When (A) makes the first move in the game, the (A)-start game balance number, $b^A_g(G)$, is the value of $d$ when both players play optimally, and when (I) makes the first move in the game, the (I)-start game balance number, $b^I_g(G)$, is the value of $d$ when both players play optimally. Among other results, we show that if $G$ has order $n$, then $-\log_2(n) \le b^A_g(G) \le \frac{n}{2}$ if $n$ is even and $0 \le b^A_g(G) \le \frac{n}{2} + \log_2(n)$ if $n$ is odd. Moreover we show that $b^A_g(G) + b^I_g(\overline{G}) = \lfloor n/2 \rfloor$., Comment: 18 pages, 1 Figure, 2 Tables

An identifying open code of a graph $G$ is a set $S$ of vertices that is both a separating open code (that is, $N_G(u) \cap S \ne N_G(v) \cap S$ for all distinct vertices $u$ and $v$ in $G$) and a total dominating set (that is, $N(v) \cap S \ne \emptyset$ for all vertices~$v$ in $G$). Such a set exists if and only if the graph $G$ is open twin-free and isolate-free; and the minimum cardinality of an identifying open code in an open twin-free and isolate-free graph $G$ is denoted by $\gamma^{{\rm {\small IOC}}}(G)$. We study the smallest size of an identifying open code of a graph, in relation with its order and its maximum degree. For $\Delta$ a fixed integer at least $3$, if $G$ is a connected graph of order $n \ge 5$ that contains no $4$-cycle and is open twin-free with maximum degree bounded above by $\Delta$, then we show that $\gamma^{{\rm {\small IOC}}}(G) \le \left( \frac{2\Delta - 1}{\Delta} \right) n$, unless $G$ is obtained from a star $K_{1,\Delta}$ by subdividing every edge exactly once. Moreover, we show that the bound is best possible by constructing graphs that reach the bound.

In network theory, the domination parameter is vital in investigating several structural features of the networks, including connectedness, their tendency to form clusters, compactness, and symmetry. In this context, various domination parameters have been created using several properties to determine where machines should be placed to ensure that all the places are monitored. To ensure efficient and effective operation, a piece of equipment must monitor their network (power networks) to answer whenever there is a change in the demand and availability conditions. Consequently, phasor measurement units (PMUs) are utilised by numerous electrical companies to monitor their networks perpetually. Overseeing an electrical system which consists of minimum PMUs is the same as the vertex covering the problem of graph theory, in which a subset D of a vertex set V is a power dominating set (PDS) if it monitors generators, cables, and all other components, in the electrical system using a few guidelines. Hypercube is one of the versatile, most popular, adaptable, and convertible interconnection networks. Its appealing qualities led to the development of other hypercube variants. A fractal cubic network is a new variant of the hypercube that can be used as a best substitute in case faults occur in the hypercube, which was wrongly defined in [Eng. Sci. Technol. 18(1) (2015) 32-41]. Arulperumjothi et al. have recently corrected this definition and redefined this variant with the exact definition in [Appl. Math. Comput. 452 (2023) 128037]. This article determines the PDS of the fractal cubic network. Further, we investigate the resolving power dominating set (RPDS), which contrasts starkly with hypercubes, where resolving power domination is inherently challenging.

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex is adjacent to a vertex in $S$. The total domination number $\gamma_t(G)$ is the minimum cardinality of a total dominating set of $G$. The total domination subdivision number $\mbox{sd}_{\gamma_t}(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to increase the total domination number. Haynes et al. (Discrete Math. 286 (2004) 195--202) have given a constructive characterization of trees whose total domination subdivision number is~$3$. In this paper, we give new characterizations of trees whose total domination subdivision number is 3., Comment: 15 pages, 7 figures

An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhood and $S$. It was conjectured that there exists a constant $c$ such that for every connected closed-twin-free graph $G$ of order $n$ and maximum degree $\Delta$, the graph $G$ admits an identifying code of size at most $\left( \frac{\Delta-1}{\Delta} \right) n+c$. In [D. Chakraborty, F. Foucaud, M. A. Henning, and T. Lehtil\"{a}. Identifying codes in graphs of given maximum degree: Characterizing trees. arXiv preprint arXiv:2403.13172, 2024], we proved the conjecture for all trees. In this article, we show that the conjecture holds for all triangle-free graphs, with the same list of exceptional graphs needing $c>0$ as for trees: for $\Delta\ge 3$, $c=1/3$ suffices and there is only a set of 12 trees requiring $c>0$ for $\Delta=3$, and when $\Delta\ge 4$ this set is reduced to the $\Delta$-star only. Our proof is by induction, whose starting point is the above result for trees. Along the way, we prove a generalized version of Bondy's theorem on induced subsets [J. A. Bondy. Induced subsets. Journal of Combinatorial Theory, Series B, 1972] that we use as a tool in our proofs. We also use our main result for triangle-free graphs, to prove the upper bound $\left( \frac{\Delta-1}{\Delta} \right) n+1/\Delta+4t$ for graphs that can be made triangle-free by the removal of $t$ edges.

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G - S$ obtained by removing all vertices in $S$ is isolate-free. The domination number $\gamma(G)$ and the restrained domination number $\gamma_{r}(G)$ are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of $G$. Let $G$ be a cubic graph of order~$n$. A classical result of Reed [Combin. Probab. Comput. 5 (1996), 277--295] states that $\gamma(G) \le \frac{3}{8}n$, and this bound is best possible. To determine a best possible upper bound on the restrained domination number of $G$ is more challenging, and we prove that $\gamma_{r}(G) \le \frac{2}{5}n$., Comment: 39 pages, 16 figures

An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a dominating set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhoods and $S$. It was conjectured that there exists an absolute constant $c$ such that for every connected graph $G$ of order $n$ and maximum degree $\Delta$, the graph $G$ admits an identifying code of size at most $( \frac{\Delta-1}{\Delta} )n +c$. We provide significant support for this conjecture by exactly characterizing every tree requiring a positive constant $c$ together with the exact value of the constant. Hence, proving the conjecture for trees. For $\Delta=2$ (the graph is a path or a cycle), it is long known that $c=3/2$ suffices. For trees, for each $\Delta\ge 3$, we show that $c=1/\Delta\le 1/3$ suffices and that $c$ is required to have a positive value only for a finite number of trees. In particular, for $\Delta = 3$, there are 12 trees with a positive constant $c$ and, for each $\Delta \ge 4$, the only tree with positive constant $c$ is the $\Delta$-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtil\"a. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022]. We remark that there are infinitely many trees for which the bound is tight when $\Delta=3$; for every $\Delta\ge 4$, we construct an infinite family of trees of order $n$ with identification number very close to the bound, namely $\left( \frac{\Delta-1+\frac{1}{\Delta-2}}{\Delta+\frac{2}{\Delta-2}} \right) n > (\frac{\Delta-1}{\Delta} ) n -\frac{n}{\Delta^2}$. Furthermore, we also give a new tight upper bound for identification number on trees by showing that the sum of the domination and identification numbers of any tree $T$ is at most its number of vertices.

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S . The domination number of G, denoted by $\gamma$(G), is the minimum cardinality of a dominating set in G. In a breakthrough paper in 2008, L{\"o}wenstein and Rautenbach proved that if G is a cubic graph of order n and girth at least 83, then $\gamma$(G) $\le$ n/3. A natural question is if this girth condition can be lowered. The question gave birth to two 1/3-conjectures for domination in cubic graphs. The first conjecture, posed by Verstraete in 2010, states that if G is a cubic graph on n vertices with girth at least 6, then $\gamma$(G) $\le$ n/3. The second conjecture, first posed as a question by Kostochka in 2009, states that if G is a cubic, bipartite graph of order n, then $\gamma$(G) $\le$n/3. In this paper, we prove Verstraete's conjecture when there is no 7-cycle and no 8-cycle, and we prove the Kostochka's related conjecture for bipartite graphs when there is no 4-cycle and no 8-cycle.

In this follow-up to [M.G.~Cornet, P.~Torres, arXiv:2308.15603], where the $k$-tuple domination number and the 2-packing number in Kneser graphs $K(n,r)$ were studied, we are concerned with two variations, the $k$-domination number, ${\gamma_{k}}(K(n,r))$, and the $k$-tuple total domination number, ${\gamma_{t\times k}}(K(n,r))$, of $K(n,r)$. For both invariants we prove monotonicity results by showing that ${\gamma_{k}}(K(n,r))\ge {\gamma_{k}}(K(n+1,r))$ holds for any $n\ge 2(k+r)$, and ${\gamma_{t\times k}}(K(n,r))\ge {\gamma_{t\times k}}(K(n+1,r))$ holds for any $n\ge 2r+1$. We prove that ${\gamma_{k}}(K(n,r))={\gamma_{t\times k}}(K(n,r))=k+r$ when $n\geq r(k+r)$, and that in this case every ${\gamma_{k}}$-set and ${\gamma_{t\times k}}$-set is a clique, while ${\gamma_{k}}(r(k+r)-1,r)={\gamma_{t\times k}}(r(k+r)-1,r)=k+r+1$, for any $k\ge 2$. Concerning the 2-packing number, $\rho_2(K(n,r))$, of $K(n,r)$, we prove the exact values of $\rho_2(K(3r-3,r))$ when $r\ge 10$, and give sufficient conditions for $\rho_2(K(n,r))$ to be equal to some small values by imposing bounds on $r$ with respect to $n$. We also prove a version of monotonicity for the $2$-packing number of Kneser graphs., Comment: 15 pages, 3 tables

While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number $\gamma_t(G)$ (resp. $\Gamma_t(G)$) of $G$. We prove that $Z(G)+\gamma_t(G)\le n(G)$ and $Z(G)+\frac{\Gamma_t(G)}{2}\le n(G)$ holds for any graph $G$ with no isolated vertices of order $n(G)$. Both bounds are sharp as demonstrated by several infinite families of graphs. In particular, we show that every graph $H$ is an induced subgraph of a graph $G$ with $Z(G)+\frac{\Gamma_t(G)}{2}=n(G)$. Furthermore, we prove a characterization of graphs with power domination equal to $1$, from which we derive a characterization of the extremal graphs attaining the trivial lower bound $Z(G)\ge \delta(G)$. The class of graphs that appears in the corresponding characterizations is obtained by extending an idea from [D.D.~Row, A technique for computing the zero forcing number of a graph with a cut-vertex, Linear Alg.\ Appl.\ 436 (2012) 4423--4432], where the graphs with zero forcing number equal to $2$ were characterized., Comment: 18 pages, 1 figure

A vertex $u$ in a graph $G$ totally dominates a vertex $v$ if $u$ is adjacent to $v$ in $G$. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex of $G$ is totally dominated by a vertex in $S$. The indicated total domination game is played on a graph $G$ by two players, Dominator and Staller, who take turns making a move. In each of his moves, Dominator indicates a vertex $v$ of the graph that has not been totally dominated in the previous moves, and Staller chooses (or selects) any vertex adjacent to $v$ that has not yet been played, and adds it to a set $D$ that is being built during the game. The game ends when every vertex is totally dominated, that is, when $D$ is a total dominating set of $G$. The goal of Dominator is to minimize the size of $D$, while Staller wants just the opposite. Providing that both players are playing optimally with respect to their goals, the size of the resulting set $D$ is the indicated total domination number of $G$, denoted by $\gamma_t^{\rm i}(G)$. In this paper we present several results on indicated total domination game. Among other results we prove that the indicated total domination number of a graph is bounded below by the well studied upper total domination number., Comment: 20 pages, 6 figures, 18 references, correction is Section 2

Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpi\'{n}ski product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. In this paper, we define the Sierpi\'{n}ski domination number as the minimum of $\gamma(G\otimes _f H)$ over all functions $f \colon V(G)\rightarrow V(H)$. The upper Sierpi\'{n}ski domination number is defined analogously as the corresponding maximum. After establishing general upper and lower bounds, we determine the upper Sierpi\'{n}ski domination number of the Sierpi\'{n}ski product of two cycles, and determine the lower Sierpi\'{n}ski domination number of the Sierpi\'{n}ski product of two cycles in half of the cases and in the other half cases restrict it to two values.

Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpi\'{n}ski product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. The Sierpi\'{n}ski metric dimension and the upper Sierpi\'{n}ski metric dimension of two graphs are determined. Closed formulas are determined for Sierpi\'{n}ski products of trees, and for Sierpi\'{n}ski products of two cycles where the second factor is a triangle. We also prove that the layers with respect to the second factor in a Sierpi\'{n}ski product graph are convex.

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider the $3$-bootstrap percolation number of grids with fixed widths. If $G$ is the cartesian product $P_3 \square P_m$ of two paths of orders~$3$ and $m$, we prove that $m(G,3)=\frac{3}{2}(m+1)-1$, when $m$ is odd, and $m(G,3)=\frac{3}{2}m +1$, when $m$ is even. Moreover if $G$ is the cartesian product $P_5 \square P_m$, we prove that $m(G,3)=2m+2$, when $m$ is odd, and $m(G,3)=2m+3$, when $m$ is even. If $G$ is the cartesian product $P_4 \square P_m$, we prove that $m(G,3)$ takes on one of two possible values, namely $m(G,3) = \lfloor \frac{5(m+1)}{3} \rfloor + 1$ or $m(G,3) = \lfloor \frac{5(m+1)}{3} \rfloor + 2$., Comment: 27 pages, 13 figures

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $V(G) \setminus S$ is adjacent to a vertex in $S$. A coalition in $G$ consists of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a dominating set but whose union $X \cup Y$ is a dominating set of $G$. Such sets $X$ and $Y$ form a coalition in $G$. A coalition partition, abbreviated $c$-partition, in $G$ is a partition $\mathcal{X} = \{X_1,\ldots,X_k\}$ of the vertex set $V(G)$ of $G$ such that for all $i \in [k]$, each set $X_i \in \mathcal{X}$ satisfies one of the following two conditions: (1) $X_i$ is a dominating set of $G$ with a single vertex, or (2) $X_i$ forms a coalition with some other set $X_j \in \mathcal{X}$. %The coalition number ${C}(G)$ is the maximum cardinality of a $c$-partition of $G$. Let ${\cal A} = \{A_1,\ldots,A_r\}$ and ${\cal B}= \{B_1,\ldots, B_s\}$ be two partitions of $V(G)$. Partition ${\cal B}$ is a refinement of partition ${\cal A}$ if every set $B_i \in {\cal B} $ is either equal to, or a proper subset of, some set $A_j \in {\cal A}$. Further if ${\cal A} \ne {\cal B}$, then ${\cal B}$ is a proper refinement of ${\cal A}$. Partition ${\cal A}$ is a minimal $c$-partition if it is not a proper refinement of another $c$-partition. Haynes et al. [AKCE Int. J. Graphs Combin. 17 (2020), no. 2, 653--659] defined the minmin coalition number $c_{\min}(G)$ of $G$ to equal the minimum order of a minimal $c$-partition of $G$. We show that $2 \le c_{\min}(G) \le n$, and we characterize graphs $G$ of order $n$ satisfying $c_{\min}(G) = n$. A polynomial-time algorithm is given to determine if $c_{\min}(G)=2$ for a given graph $G$. A necessary and sufficient condition for a graph $G$ to satisfy $c_{\min}(G) \ge 3$ is given, and a characterization of graphs $G$ with minimum degree~$2$ and $c_{\min}(G)= 4$ is provided.

Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. For a graph $G$ of order $n$ and maximum degree at most $\Delta$, we show $Mo(G)\leq \frac{\Delta}{2}n^2-(1-o(1))c_{\Delta}n\log(\log(n)),$ where $c_{\Delta}>0$ only depends on $\Delta$ and the $o(1)$ term only depends on $n$. Furthermore, for integers $n_0$ and $\Delta$ at least $3$, we show the existence of a $\Delta$-regular graph of order $n$ at least $n_0$ with $Mo(G)\geq \frac{\Delta}{2}n^2-c'_{\Delta}n\log(n),$ where $c'_{\Delta}>0$ only depends on $\Delta$.

A set $D$ of vertices is a strong dominating set in a graph $G$, if for every vertex $x\in V(G) \setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x) \leq deg(y)$. The strong domination number $\gamma_{st}(G)$ of $G$ is the minimum cardinality of a strong dominating set in $G$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identifying these two vertices, and thereafter continuing in this manner inductively. The graphs $G_1,\ldots ,G_k$ are the primary subgraphs of $G$. In this paper, we study the strong domination number of $K_r$-gluing of two graphs and investigate the strong domination number for some particular cases of graphs from their primary subgraphs., Comment: 20 pages, 13 figures

Let $G$ be graph with vertex set $V$ and order $n=|V|$. A coalition in $G$ is a combination of two distinct sets, $A\subseteq V$ and $B\subseteq V$, which are disjoint and are not dominating sets of $G$, but $A\cup B$ is a dominating set of $G$. A coalition partition of $G$ is a partition $\mathcal{P}=\{S_1,\ldots,S_k\}$ of its vertex set $V$, where each set $S_i\in \mathcal{P}$ is either a dominating set of $G$ with only one vertex, or it is not a dominating set but forms a coalition with some other set $S_j \in \mathcal{P}$. The coalition number $C(G)$ is the maximum cardinality of a coalition partition of $G$. To represent a coalition partition $\mathcal{P}$ of $G$, a coalition graph $\CG(G, \mathcal{P})$ is created, where each vertex of the graph corresponds to a member of $\mathcal{P}$ and two vertices are adjacent if and only if their corresponding sets form a coalition in $G$. A coalition partition $\mathcal{P}$ of $G$ is a singleton coalition partition if every set in $\mathcal{P}$ consists of a single vertex. If a graph $G$ has a singleton coalition partition, then $G$ is referred to as a singleton-partition graph. A graph $H$ is called a singleton coalition graph of a graph $G$ if there exists a singleton coalition partition $\mathcal{P}$ of $G$ such that the coalition graph $\CG(G,\mathcal{P})$ is isomorphic to $H$. A singleton coalition graph chain with an initial graph $G_1$ is defined as the sequence $G_1\rightarrow G_2\rightarrow G_3\rightarrow\cdots$ where all graphs $G_i$ are singleton-partition graphs, and $\CG(G_i,\Gamma_1)=G_{i+1}$, where $\Gamma_1$ represents a singleton coalition partition of $G_i$. In this paper, we address two open problems posed by Haynes et al. We characterize all graphs $G$ of order $n$ and minimum degree $\delta(G)=2$ such that $C(G)=n$ and investigate the singleton coalition graph chain starting with graphs $G$ where $\delta(G)\le 2$.

Let $G=(V,E)$ be a graph with no isolated vertices. A vertex $v$ totally dominate a vertex $w$ ($w \ne v$), if $v$ is adjacent to $w$. A set $D \subseteq V$ called a total dominating set of $G$ if every vertex $v\in V$ is totally dominated by some vertex in $D$. The minimum cardinality of a total dominating set is the total domination number of $G$ and is denoted by $\gamma_t(G)$. A total dominator coloring of graph $G$ is a proper coloring of vertices of $G$, so that each vertex totally dominates some color class. The total dominator chromatic number $\chi_{td}(G)$ of $G$ is the least number of colors required for a total dominator coloring of $G$. The Total Dominator Coloring problem is to find a total dominator coloring of $G$ using the minimum number of colors. It is known that the decision version of this problem is NP-complete for general graphs. We show that it remains NP-complete even when restricted to bipartite, planar and split graphs. We further study the Total Dominator Coloring problem for various graph classes, including trees, cographs and chain graphs. First, we characterize the trees having $\chi_{td}(T)=\gamma_t(T)+1$, which completes the characterization of trees achieving all possible values of $\chi_{td}(T)$. Also, we show that for a cograph $G$, $\chi_{td}(G)$ can be computed in linear-time. Moreover, we show that $2 \le \chi_{td}(G) \le 4$ for a chain graph $G$ and give characterization of chain graphs for every possible value of $\chi_{td}(G)$ in linear-time., Comment: V1, 18 pages, 1 figure

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$ have distinct neighborhoods within $S$. The smallest size of such a set is denoted by $\gamma^L_t(G)$. It has been conjectured that $\gamma^L_t(G)\leq\frac{2n}{3}$ holds for every twin-free graph $G$ of order $n$ without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.

A graph is $\alpha$-excellent if every vertex of the graph is contained in some maximum independent set of the graph. In this paper, we present two characterizations of the $\alpha$-excellent $2$-trees., Comment: 10 pages, 3 figures

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. Two vertices are neighbors if they are adjacent. A super dominating set is a dominating set $S$ with the additional property that every vertex in $V \setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in $V \setminus S$. Moreover if every vertex in $V \setminus S$ has degree at least~$2$, then $S$ is an end super dominating set. The end super domination number is the minimum cardinality of an end super dominating set. We give applications of end super dominating sets as main servers and temporary servers of networks. We determine the exact value of the end super domination number for specific classes of graphs, and we count the number of end super dominating sets in these graphs. Tight upper bounds on the end super domination number are established, where the graph is modified by vertex (edge) removal and contraction., Comment: 25 pages, 11 figures

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The common independence number $\alpha_c(G)$ of $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set of cardinality at least~$r$. The common independence number is squeezed between the independent domination number $i(G)$ and the independence number $\alpha(G)$ of $G$, that is, $\gamma(G) \le i(G) \le \alpha_c(G) \le \alpha(G)$. A graph $G$ is domination perfect if $\gamma(H) = i(H)$ for every induced subgraph $H$ of $G$. We define a graph $G$ as common domination perfect if $\gamma(H) = \alpha_c(H)$ for every induced subgraph $H$ of $G$. We provide a characterization of common domination perfect graphs in terms of ten forbidden induced subgraphs., Comment: 11 pages, 2 figures

Let $G$ be a graph with vertex set $V$ and of order $n = |V|$, and let $\delta(G)$ and $\Delta(G)$ be the minimum and maximum degree of $G$, respectively. Two disjoint sets $V_1, V_2 \subseteq V$ form a coalition in $G$ if none of them is a dominating set of $G$ but their union $V_1\cup V_2$ is. A vertex partition $\Psi=\{V_1,\ldots, V_k\}$ of $V$ is a coalition partition of $G$ if every set $V_i\in \Psi$ is either a dominating set of $G$ with the cardinality $|V_i|=1$, or is not a dominating set but for some $V_j\in \Psi$, $V_i$ and $V_j$ form a coalition. The maximum cardinality of a coalition partition of $G$ is the coalition number $\mathcal{C}(G)$ of $G$. Given a coalition partition $\Psi = \{V_1, \ldots, V_k\}$ of $G$, a coalition graph $\CG(G, \Psi)$ is associated on $\Psi$ such that there is a one-to-one correspondence between its vertices and the members of $\Psi$, where two vertices of $\CG(G, \Psi)$ are adjacent if and only if the corresponding sets form a coalition in $G$. In this paper, we partially solve one of the open problems posed in Haynes et al. \cite{coal0} and we solve two open problems posed by Haynes et al. \cite{coal1}. We characterize all graphs $G$ with $\delta(G) \le 1$ and $\mathcal{C}(G)=n$, and we characterize all trees $T$ with $\mathcal{C}(T)=n-1$. We determine the number of coalition graphs that can be defined by all coalition partitions of a given path. Furthermore, we show that there is no universal coalition path, a path whose coalition partitions defines all possible coalition graphs.