在本文中，我们继续研究图中的全控制对策[Graphs Combin. 31（5）（2015），1453- 1462]，其中玩家Dominator和Staller交替地选择$G$的顶点。每个顶点的选择必须严格增加完全支配的顶点数，其中一个顶点完全支配另一个顶点，如果它们是邻居。该过程最终产生$G$的全支配集$S$，其中每个顶点完全被$S$中的顶点支配。$G$的博弈总控制数$γ_{\rm tg}(G)$（分别为Staller-start game total dominance number，$γ_{\rm tg}'(G)$）是当Dominator（分别为Staller）开始博弈且双方都处于最优状态时所选择的顶点数。对于一般的图$G$，有时候是$γ_{\rm tg}(G) > γ_{\rm tg}'(G)$。证明了如果$G$是一个没有孤立点的森林，则$γ_{\rm tg}(G) \le γ_{\rm tg}'(G)$ .利用这个结果，我们刻画了具有相等全控制数和对策全控制数的树。
英文摘要：
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 total domination number, $γ_{\rm tg}(G)$, (respectively, Staller-start game total domination number, $γ_{\rm tg}'(G)$) of $G$ is the number of vertices chosen when Dominator (respectively, Staller) starts the game and both players play optimally. For general graphs $G$, sometimes $γ_{\rm tg}(G) > γ_{\rm tg}'(G)$. We show that if $G$ is a forest with no isolated vertex, then $γ_{\rm tg}(G) \le γ_{\rm tg}'(G)$. Using this result, we characterize the trees with equal total domination and game total domination number.
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.
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…
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…
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.
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.
The 2-domination number $γ_2(G)$ of a graph $G$ is the minimum cardinality of a set $ D \subseteq V(G) $ for which every vertex outside $ D $ is adjacent to at least two vertices in $ D $. Clearly, $ γ_2(G) $ cannot be smaller than the domination number $ γ(G) $. We consider a large class of graphs and characterize those members which satisfy $γ_2=γ$. For the general case, we prove that it is NP-hard to decide whether $γ_2=γ$ holds. We also give a necessary and sufficient condition for a graph to satisfy the equality hereditarily.
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…
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 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…
The Grundy domination number, ${γ_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $j<i$. It is well known that the Grundy domination number of any graph $G$ is greater than or equal to the upper domination number $Γ(G)$, which is in turn greater than or equal to the independence number $α(G)$. In this paper, we initiate the study of the class of graphs $G$ with $Γ(G)={γ_{\rm gr…
The minimum degree question for the Maker Breaker Domination Game
Johannes Kepler University Linz, Institute for Algebra（林茨约翰·开普勒大学代数研究所）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉登计算与应用数学研究所）；Institute of Science and Technology, Austria（奥地利科学技术研究所）
