Let $γ_g(G)$ and $γ_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $γ_g$-perfect (resp. $γ_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $γ_g(F)=γ(F)$ (resp. $γ_{tg}(F)=γ_t(F)$). A recursive characterization of $γ_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $γ_g$-perfect graphs. It is proved that every minimally $γ_g$-imperfect graph has domination number $2$. All minimally $γ_g$-imperfect triangle-free graphs are determined. It is also proved that $γ_{tg}$-perfect graphs are precisely $\overline{2P_3}$-free cographs.
Two new techniques are introduced into the theory of the domination game. The cutting lemma bounds the game domination number of a partially dominated graph with the game domination number of suitably modified partially dominated graph. The union lemma bounds the S-game domination number of a disjoint union of paths using appropriate weighting functions. Using these tools a conjecture asserting that the so-called three legged spiders are game domination critical graphs is proved. An extended cutting lemma is also derived and all game domination critical trees on 18, 19, and 20 vertices are lis…
Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or…
The connected domination game was recently introduced by Borowiecki, Fiedorowicz and Sidorowicz as another variation of the domination game. The rules are essentially the same, except that the set of played vertices must be connected at all stages of the game. We answer a problem from their paper regarding the relation between the number of moves in a game where Dominator/Staller starts the game. In this paper we also study the relation to the diameter and present graphs with small connected game domination number. We determine the values on the lexicographic product graphs, and consider the e…
The domination game is played on a graph G. Vertices are chosen, one at a time, by two players Dominator and Staller. Each chosen vertex must enlarge the set of vertices of G dominated to that point in the game. Both players use an optimal strategy---Dominator plays so as to end the game as quickly as possible while Staller plays in such a way that the game lasts as many steps as possible. The game domination number of G is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number of G when Staller starts the game. In this paper these two games a…
The Z-domination game is a variant of the domination game in which each newly selected vertex $u$ in the game must have a not yet dominated neighbor, but after the move all vertices from the closed neighborhood of $u$ are declared to be dominated. The Z-domination game is the fastest among the five natural domination games. The corresponding game Z-domination number of a graph $G$ is denoted by $γ_{Zg}(G)$. It is proved that the game domination number and the game total domination number of a graph can be expressed as the game Z-domination number of appropriate lexicographic products. Graphs w…
In this paper we describe a strategy for Dominator that finishes the total domination game in at most $3/4n$ moves for every graph $G$ on $n$ vertices without any isolated vertices or edges, confirming the 3/4-conjecture for the total domination game made by Henning, Klav{ž}ar, and Rall.
A \emph{perfect Roman dominating function} (PRDF) on a graph $G = (V, E)$ is a function $f : V \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The weight of a PRDF is the value $w(f) = \sum_{u \in V}f(u)$. The minimum weight of a PRDF on a graph $G$ is called the \emph{perfect Roman domination number $γ_R^p(G)$} of $G$. A graph $G$ is perfect Roman domination domination stable if the perfect Roman domination number of $G$ remains unchanged under the removal of any vertex. In this paper, we …
In this paper, an upper bound for the perfect Italian domination number of the cartesian product of any two graphs is obtained and the exact value of this parameter for cartesian product of some special graphs are obtained. We have also proved that for any two positive integers $a$, $b$ there exists a graph $G$ and an induced subgraph $H$ of $G$ such that $γ_I^p(G) = a$ and $γ_I^p(H) = b$. Relationship of the perfect Italian domination number with the Roman domination number and the perfect domination number of a graph $G$ are obtained and the corresponding realization problems are also solved…
In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set $S$ of $G$ in which every vertex is totally dominated by a vertex in $S$. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game …
In this paper a relationship is established between the domination game and minimal edge cuts. It is proved that the game domination number of a connected graph can be bounded above in terms of the size of minimal edge cuts. In particular, if $C$ a minimum edge cut of a connected graph $G$, then $γ_g(G) \le γ_g(G\setminus C) + 2κ'(G)$. Double-Staller graphs are introduced in order to show that this upper bound can be attained for graphs with a bridge. The obtained results are used to extend the family of known traceable graphs whose game domination numbers are at most one-half their order. Alo…
Let $γ(G)$ and $β(G)$ denote the domination number and the covering number of a graph $G$, respectively. A connected non-trivial graph $G$ is said to be $γβ$-{perfect} if $γ(H)=β(H)$ for every non-trivial induced connected subgraph $H$ of $G$. In this note we present an elementary proof of a characterization of the $γβ$-perfect graphs.
The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a function $f : V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $v$ with $f(v)=0$ is adjacent to exactly one vertex $u$ for which $f(u)=2$. The weight of a perfect Roman dominating function $f$ is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on $G$. In this…
The game total domination number, ${γ_{g}^{t}}$, was introduced by Henning et al.\ in 2015. In this paper we study the effect of vertex predomination on the game total domination number. We prove that ${γ_{g}^{t}}(G|v) \geq {γ_{g}^{t}}(G) - 2$ holds for all vertices $v$ of a graph $G$ and present infinite families attaining the equality. To achieve this, some new variations of the total domination game are introduced. The effect of vertex removal is also studied. We show that ${γ_{g}^{t}}(G) \leq {γ_{g}^{t}}(G-v) + 4$ and ${γ_{g}^{t}}'(G) \leq {γ_{g}^{t}}'(G-v) + 4$.
It is conjectured that the game domination number is at most $3n/5$ for every $n$-vertex graph which does not contain isolated vertices. It was proved in the recent years that the conjecture holds for several graph classes, including the class of forests and that of graphs with minimum degree at least two. Here we prove that the slightly bigger upper bound $5n/8$ is valid for every isolate-free graph.
Maker-Breaker total domination game in graphs is introduced as a natural counterpart to the Maker-Breaker domination game recently studied by Duchêne, 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 infinit…
The $1/2$-conjecture on the domination game asserts that if $G$ is a traceable graph, then the game domination number $γ_g(G)$ of $G$ is at most $\left\lceil \frac{n(G)}{2} \right\rceil$. A traceable graph is a $1/2$-graph if $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ holds. It is proved that the so-called hatted cycles are $1/2$-graphs and that unicyclic graphs fulfill the $1/2$-conjecture. Several additional families of graphs that support the conjecture are determined and computer experiments related to the conjecture described.
The domination game is played on a graph $G$ by two players, Dominator and Staller, who alternate in selecting vertices until each vertex in the graph $G$ is contained in the closed neighbourhood of the set of selected vertices. Dominator's aim is to reach this state in as few moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we prove that if $G$ has $n$ vertices and minimum degree at least 2, then Dominator has a strategy to finish the domination game on $G$ within $10n/17+1/17$ moves, thus making progress towards a conjecture by Bujt{á}s, Iršič and…
In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph $\mathcal{H}$ by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices dominated so far. The game is over if all vertices of $\mathcal{H}$ are dominated. Dominator aims to finish the game as soon as possible, while Staller aims to delay the end of the game. If each player plays optimally and Dominator starts, the length of the game is the invariant `game domination number' denoted by $γ_g(\mathcal{H})$. This def…
For a graph $G = (V,E),$ a subset $S$ of $V$ is a perfect dominating set of $G$ if every vertex not in $S$ is adjacent to exactly one vertex in $S.$ The perfect domination number, $γ_p(G),$ is the minimum cardinality of a perfect dominating set of $G.$ The perfect domination number is found for knights graphs on square, rectangular, and infinite chessboards. Indeed, exact values or bounds are given for all chessboards except those with 3 rows and number of columns congruent to 1, 2, or 3 modulo 8.
An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V}f(v) $ and the Italian domination number, $ γ_I(G) $, is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on $G$, if for every vertex $ v \in V(G) $ with $ f(v) = 0 $ the total weight assigned by $f$ to the neighbours of $ v $ is exactly 2, i.e., all the neighbours of $u$ are assigned the weight 0 by $f…
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 $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The common independence number $α_c(G)$ 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 $α(G)$ of $G$, that is, $γ(G) \le i(G) \le α_c(G) \le α(G)$. A graph $G$ is dominat…
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 ($γ_{\rm tcg}(G)$) $γ_{\rm cg}(G)$ of $G$. We show that $γ_{\rm tcg}(G)\in \{γ_{\rm cg}(G), γ_{\rm cg}(G) + 1, γ_{\rm cg}(G) + 2\}$, and consequently define $G$ as …
The domination game is an optimization game played by two players, Dominator and Staller, who alternately select vertices in a graph $G$. A vertex is said to be dominated if it has been selected or is adjacent to a selected vertex. Each selected vertex must strictly increase the number of dominated vertices at the time of its selection, and the game ends once every vertex in $G$ is dominated. Dominator aims to keep the game as short as possible, while Staller tries to achieve the opposite. In this article, we prove that for any graph $G$ on $n$ vertices, Dominator has a strategy to end the gam…
The connected domination game is played just as the domination game, with an additional requirement that at each stage of the game the vertices played induce a connected subgraph. The number of moves in a D-game (an S-game, resp.) on a graph $G$ when both players play optimally is denoted by $γ_{\rm cg}(G)$ ($γ_{\rm cg}'(G)$, resp.). Connected Game Continuation Principle is established as a substitute for the classical Continuation Principle which does not hold for the connected domination game. Let $G|x$ denote the graph $G$ together with a declaration that the vertex $x$ is already dominated…
We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the $(2:1)$ biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted variant of the non-biased game we prove that the maker can win on every graph with…
The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph $G$. Each vertex chosen must strictly increase the number of vertices totally dominated. This process eventually produces a total dominating set of $G$. Dominator wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game total domination number of $G$, $γ_{\rm tg}(G)$, is the number of vertices chosen when Dominator starts the game and both players play optimally. Recently, Henni…
Let $γ_g(G)$ be the game domination number of a graph $G$. It is proved that if ${\rm diam}(G) = 2$, then $γ_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil- \left\lfloor \frac{n(G)}{11}\right\rfloor$. The bound is attained: if ${\rm diam}(G) = 2$ and $n(G) \le 10$, then $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ if and only if $G$ is one of seven sporadic graphs with $n(G)\le 6$ or the Petersen graph, and there are exactly ten graphs of diameter $2$ and order $11$ that attain the bound.
The domination game is played on a graph $G$ by two players, named Dominator and Staller. They alternatively select vertices of $G$ such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator's goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Game 2 are variants of the game in which Dominator and Staller has the first move, respectively. The game domination number $γ_g(G)$, and the Staller-start game domination number $γ_g'(G)$, is the numb…
In the domination game, introduced by Brešar, Klavžar and Rall in 2010, Dominator and Staller alternately select a vertex of a graph $G$. A move is legal if the selected vertex $v$ dominates at least one new vertex -- that is, if we have a $u\in N[v]$ for which no vertex from $N[u]$ was chosen up to this point of the game. The game ends when no more legal moves can be made, and its length equals the number of vertices selected. The goal of Dominator is to minimize whilst that of Staller is to maximize the length of the game. The game domination number $γ_g(G)$ of $G$ is the length of the domin…
A $k-$quasiperfect dominating set ($k\ge 1$) of a graph $G$ is a vertex subset $S$ such that every vertex not in $S$ is adjacent to at least one and at most k vertices in $S$. The cardinality of a minimum k-quasiperfect dominating set in $G$ is denoted by $γ_{\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept. The quasiperfect domination chain $γ_{\stackrel{}{11}}(G)\geγ_{\stackrel{}{12}}(G)\ge\dots\geγ_{\stackrel{}{1Δ}}(G)=γ(G)$, indicates what it is lost in size when you move towards a more perfect domination…
In the domination game studied here, Dominator and Staller alternately choose a vertex of a graph $G$ and take it into a set $D$. The number of vertices dominated by the set $D$ must increase in each single turn and the game ends when $D$ becomes a dominating set of $G$. Dominator aims to minimize whilst Staller aims to maximize the number of turns (or equivalently, the size of the dominating set $D$ obtained at the end). Assuming that Dominator starts and both players play optimally, the number of turns is called the game domination number $γ_g(G)$ of $G$. Kinnersley, West and Zamani verified…
In the total domination game played on a graph $G$, players Dominator and Staller alternately select vertices of $G$, as long as possible, such that each vertex chosen increases the number of vertices totally dominated. Dominator (Staller) wishes to minimize (maximize) the number of vertices selected. The game total domination number, $γ_{\rm tg}(G)$, of $G$ is the number of vertices chosen when Dominator starts the game and both players play optimally. If a vertex $v$ of $G$ is declared to be already totally dominated, then we denote this graph by $G|v$. In this paper the total domination gam…
In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set $S$ of $G$ in which every vertex is totally dominated by a vertex in $S$. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game …
The game domination number is a graph invariant that arises from a game, which is related to graph domination in a similar way as the game chromatic number is related to graph coloring. In this paper we show that verifying whether the game domination number of a graph is bounded by a given integer is PSPACE-complete. This contrasts the situation of the game coloring problem whose complexity is still unknown.
In a hypergraph with vertex set $V$ and edge set $E$, a real-valued function $f: V \to [0, 1]$ is a fractional transversal if $\sum_{v\in e} f(v) \ge 1$ for every edge $e \in E$. Its size is $|f| := \sum_{v \in V} f(v)$, and the fractional transversal number is the smallest possible $|f|$. We consider a game scenario where two players with opposite goals construct a fractional transversal incrementally, trying to minimize and maximize $|f|$, respectively. We prove that both players have strategies to achieve their common optimum, and they can reach their goals using rational weights.
Let $γ_g(G)$ be the game domination number of a graph $G$. Rall conjectured that if $G$ is a traceable graph, then $γ_g(G) \le \left\lceil \frac{1}{2}n(G)\right\rceil$. Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if $δ(G) \geq 2$, then $γ_g(G) \leq \left\lceil \frac{1}{2}n(G) \right\rceil$. We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound $γ_g(G) \le \left\lceil \frac{11}{20} \, n(G) \right\rceil$ over the class of claw-free graphs of minimum degree at least…
We show there is an uncountable number of parallel total perfect codes in the integer lattice graph $Λ$ of $\R^2$. In contrast, there is just one 1-perfect code in $Λ$ and one total perfect code in $Λ$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of $Λ$ and $C_m\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\Z_{2d^2+2d+1}$ with generator set $\{1,2d^…
A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect dominating set in $G$ is denoted by $γ_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge γ_ {\stackrel{}{11}}(G) \ge γ_ {\stackrel{}{12}}(G)\ge \ldots \ge γ_ {\stackrel{}{1Δ}}(G)=γ(G)$ in order…
In the $(a,b)$-biased Maker-Breaker domination game, two players alternately select unplayed vertices in a graph $G$ such that Dominator selects $a$ and Staller selects $b$ vertices per move. Dominator wins if the vertices he selected during the game form a dominating set of $G$, while Staller wins if she can prevent Dominator from achieving this goal. Given a positive integer $b$, Dominator's threshold, $\textrm{a}_b$, is the minimum $a$ such that Dominator wins the $(a,b)$-biased game on $G$ when he starts the game. Similarly, $\textrm{a}'_b$ denotes the minimum $a$ such that Dominator wins …
The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $γ_{\rm MB}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denot…
We introduce the Maker-Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, select a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker-Breaker games, that is studied here in a combinatorial context. In this paper, we first prove that deciding the winner of the Maker-Breaker domination game is PSPACE-complete, even for bipartite graphs and split…
Perfect codes in the $n$-dimensio\-nal grid $Λ_n$ of the lattice $\mathbb{Z}^n$ ($0<n\in\mathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance in $\mathbb{Z}^n$ given between $u=(u_1,\cdots,u_n)$ and $v=(v_1, \ldots,v_n)$ as the graph distance $h(u,v)$ in $Λ_n$, if $|u_i-v_i|\le 1$, for all $i\in\{1, \ldots,n\}$, and as $n+1$, otherwise. Such codes are extended to multilattice graphs $Γ_n$ obtained by glueing ternary $n$-cubes along their codimension 1 ternary subcubes in such a way that each binary $n$-subcube is contained in a unique maximal lattice of $Γ_n$. T…
The Maker-Breaker domination game is played on a graph $G$ by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex $v \in V(G)$. Dominator's fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant $γ'_{\rm SMB}(G)$ (resp., $γ_{\rm SMB}(G)$) which is the smallest integer $k$ such that, under any str…
