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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.

We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such a coloring. We show that this notion generalizes the dominator and total dominator chromatic numbers, and provide some general bounds and algorithmic results. We also investigate the specific cases where $P$ is that the subset contains at least one edge or that the subset is connected.

TxGraffiti is an automated conjecturing program that produces graph theoretic conjectures in the form of conjectured inequalities. This program written and maintained by the second author since 2017 was inspired by the successes of previous automated conjecturing programs including Fajtlowicz's GRAFFITI and DeLaVi\~{n}a's GRAFFITI.pc. In this paper we prove and generalize several conjectures generated by TxGraffiti when it was prompted to conjecture on the \emph{independence number}, the \emph{domination number}, and the \emph{matching number} (and generalizations of each of these graph invariants). Moreover, in several instances we also show the proposed inequalities relating these graph invariants are sharp.

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G\setminus D$ has a neighbor in $D$, while $D$ is a 2-dominating set of $G$ if every vertex belonging to $V_G\setminus D$ is joined by at least two edges with a vertex or vertices in $D$. A graph $G$ is a $(2,2)$-dominated graph if it has a pair $(D,D')$ of disjoint $2$-dominating sets of vertices of $G$. In this paper we present two characterizations of minimal $(2,2)$-dominated graphs., Comment: 8 pages, 4 figures

A graph $G$ is a DTDP-graph if it has a pair $(D,T)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $T$ is a total dominating set of $G$. Such graphs were studied in a number of research papers. In this paper we study further properties of DTDP-graphs and, in particular, we characterize minimal DTDP-gaphs without loops., Comment: 23 pages, 12 figures

The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($\gamma_{\rm tcg}(G)$) $\gamma_{\rm cg}(G)$ of $G$. We show that $\gamma_{\rm tcg}(G)\in \{\gamma_{\rm cg}(G), \gamma_{\rm cg}(G) + 1, \gamma_{\rm cg}(G) + 2\}$, and consequently define $G$ as Class $i$ if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i$ for $i \in \{0,1,2\}$. A large family of Class $0$ graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least $2$. We show that no tree is Class $2$ and characterize Class $1$ trees. We provide an infinite family of Class $2$ bipartite graphs., Comment: 12 pages, 2 figures

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

For a subset $S$ of vertices in a graph $G$, a vertex $v \in S$ is an enclave of $S$ if $v$ and all of its neighbors are in $S$, where a neighbor of $v$ is a vertex adjacent to $v$. A set $S$ is enclaveless if it does not contain any enclaves. The enclaveless number $\Psi(G)$ of $G$ is the maximum cardinality of an enclaveless set in $G$. As first observed in 1997 by Slater [J. Res. Nat. Bur. Standards 82 (1977), 197--202], if $G$ is a graph with $n$ vertices, then $\gamma(G) + \Psi(G) = n$ where $\gamma(G)$ is the well-studied domination number of $G$. In this paper, we continue the study of the competition-enclaveless game introduced in 2001 by Phillips and Slater [Graph Theory Notes N. Y. 41 (2001), 37--41] and defined as follows. Two players take turns in constructing a maximal enclaveless set $S$, where one player, Maximizer, tries to maximize $|S|$ and one player, Minimizer, tries to minimize~$|S|$. The competition-enclaveless game number $\Psi_g^+(G)$ of $G$ is the number of vertices played when Maximizer starts the game and both players play optimally. We study among other problems the conjecture that if $G$ is an isolate-free graph of order $n$, then $\Psi_g^+(G) \ge \frac{1}{2}n$. We prove this conjecture for regular graphs and for claw-free graphs., Comment: 15 pages, 1 figure, 23 references

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. In 2013 Goddard and Henning [Discrete Math 313 (2013), 839--854] conjectured that if $G$ is a connected cubic graph of order $n$, then $i(G) \le \frac{3}{8}n$, except if $G$ is the complete bipartite graph $K_{3,3}$ or the $5$-prism $C_5 \, \Box \, K_2$. Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. They remark that perhaps it is even true that for $n > 10$ these two families are only families for which equality holds. In this paper, we provide a new family of connected cubic graphs $G$ of order $n$ such that $i(G) = \frac{3}{8}n$. We also show that if $G$ is a subcubic graph of order $n$ with no isolated vertex, then $i(G) \le \frac{1}{2}n$, and we characterize the graphs achieving equality in this bound., Comment: Submitted to Discrete Applied Mathematics Journal, 08 Jan 2020

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a~neighbor in $D$, while $D$ is a paired-dominating set of $G$ if $D$ is a~dominating set and the subgraph induced by $D$ contains a perfect matching. A graph $G$ is a $D\!P\!D\!P$-graph if it has a pair $(D,P)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $P$ is a paired-dominating set of $G$. The study of the $D\!P\!D\!P$-graphs was initiated by Southey and Henning (Cent. Eur. J. Math. 8 (2010) 459--467; J. Comb. Optim. 22 (2011) 217--234). In this paper, we provide conditions which ensure that a graph is a $D\!P\!D\!P$-graph. In particular, we characterize the minimal $D\!P\!D\!P$-graphs., Comment: 4 figures

We propose the conjecture that the domination number $\gamma(G)$ of a $\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge domination number $\gamma_e(G)$, which coincides with the domination number of its line graph. We prove that $\gamma(G)\leq \left(1+\frac{2(\Delta-1)}{\Delta 2^{\Delta}}\right)\gamma_e(G)$ for general $\Delta\geq 1$, and $\gamma(G)\leq \left(\frac{7}{6}-\frac{1}{204}\right)\gamma_e(G)$ for $\Delta=3$. Furthermore, we verify our conjecture for cubic claw-free graphs.

A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex outside $S$ is adjacent to a vertex in $S$. A connected dominating set in $G$ is a dominating set $S$ such that the subgraph $G[S]$ induced by $S$ is connected. The connected domination number of $G$, $\gamma_c(G)$, is the minimum cardinality of a connected dominating set of $G$. A graph $G$ is said to be $k$-$\gamma_{c}$-critical if the connected domination number $\gamma_{c}(G)$ is equal to $k$ and $\gamma_{c}(G + uv) < k$ for every pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\zeta$ be the number of cut-vertices of $G$. It is known that if $G$ is a $k$-$\gamma_{c}$-critical graph, then $G$ has at most $k - 2$ cut-vertices, that is $\zeta \le k - 2$. In this paper, for $k \ge 4$ and $0 \le \zeta \le k - 2$, we show that every $k$-$\gamma_{c}$-critical graph with $\zeta$ cut-vertices has a hamiltonian path if and only if $k - 3 \le \zeta \le k - 2$., Comment: 26 pages

The edge domination number $\gamma_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on $\gamma_e(G)$ for regular and non-regular graphs $G$ in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs.

For a graph $G=(V,E)$ with no isolated vertices, a set $D\subseteq V$ is called a semipaired dominating set of G if $(i)$ $D$ is a dominating set of $G$, and $(ii)$ $D$ can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of $G$ is called the semipaired domination number of $G$, and is denoted by $\gamma_{pr2}(G)$. The \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. In this paper, we initiate the algorithmic study of the \textsc{Minimum Semipaired Domination} problem. We show that the decision version of the \textsc{Minimum Semipaired Domination} problem is NP-complete for bipartite graphs and split graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs and trees. We also propose a $1+\ln(2\Delta+2)$-approximation algorithm for the \textsc{Minimum Semipaired Domination} problem, where $\Delta$ denote the maximum degree of the graph and show that the \textsc{Minimum Semipaired Domination} problem cannot be approximated within $(1-\epsilon) \ln|V|$ for any $\epsilon > 0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log\log|V|)})$., Comment: arXiv admin note: text overlap with arXiv:1711.10891

Maker-Breaker total domination game in graphs is introduced as a natural counterpart to the Maker-Breaker domination game recently studied by Duch\^ene, Gledel, Parreau, and Renault. Both games are instances of the combinatorial Maker-Breaker games. The Maker-Breaker total domination game is played on a graph $G$ by two players who alternately take turns choosing vertices of $G$. The first player, Dominator, selects a vertex in order to totally dominate $G$ while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. It is shown that there are infinitely many connected cubic graphs in which Staller wins and that no minimum degree condition is sufficient to guarantee that Dominator wins when Staller starts the game. An amalgamation lemma is established and used to determine the outcome of the game played on grids. Cacti are also classified with respect to the outcome of the game. A connection between the game and hypergraphs is established. It is proved that the game is PSPACE-complete on split and bipartite graphs. Several problems and questions are also posed., Comment: 21 pages, 5 figures