On the structure of dominating graphs

The $k$-dominating graph $D_k(G)$ of a graph $G$ is defined on the vertex set consisting of dominating sets of $G$ with cardinality at most $k$, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to $D_k(G)$ for some graph $G$ and some positive integer $k$. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if $G$ is such a graph of order $n\ge 2$ and with $G\cong D_k(G)$, then $k=2$ and $G=K_{1,n-1}$ for some $n\ge 4$. It is also proved that for a given $r$ there exist only a finite number of $r$-regular, connected dominating graphs of connected graphs. In particular, $C_6$ and $C_8$ are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.Comment: 8 pages, 1 figur

Uniform Length Dominating Sequence Graphs

A sequence of vertices $(v_1,\, \dots , \,v_k)$ of a graph $G$ is called a {\it dominating closed neighborhood sequence} if $\{v_1,\, \dots , \,v_k\}$ is a dominating set of $G$ and $N[v_i]\nsubseteq \cup _{j=1}^{i-1} N[v_j]$ for every $i$. A graph $G$ is said to be {\it $k-$uniform} if all dominating closed neighborhood sequences have equal length $k$. Bre{\v s}ar et al. (2014) characterized $k$-uniform graphs with $k\leq 3$. In this article we extend their work by giving a complete characterization of all $k$-uniform graphs with $k\geq 4$.Comment: 7 page

Domination games played on line graphs of complete multipartite graphs

The domination game on a graph $G$ (introduced by B. Bre\v{s}ar, S. Klav\v{z}ar, D.F. Rall \cite{BKR2010}) consists of two players, Dominator and Staller, who take turns choosing a vertex from $G$ such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible, and Staller wishes to delay this process as much as possible. The game domination number $\gamma _{{\small g}}(G)$ is the number of vertices chosen when Dominator starts the game; when Staller starts, it is denoted by $\gamma _{{\small g}}^{\prime }(G).$ In this paper, the domination game on line graph $L\left( K_{\overline{m}}\right)$ of complete multipartite graph $K_{\overline{m}}$ $(\overline{m}\equiv (m_{1},...,m_{n})\in \mathbb{N} ^{n})$ is considered, the exact values for game domination numbers are obtained and optimal strategy for both players is described. Particularly, it is proved that for $m_{1}\leq m_{2}\leq ...\leq m_{n}$ both $\gamma _{{\small g}}\left( L\left( K_{\overline{m}}\right) \right) =\min \left\{ \left\lceil \frac{2}{3}\left\vert V\left( K_{\overline{m}}\right) \right\vert \right\rceil ,\right.$ $\left. 2\max \left\{ \left\lceil \frac{1}{2}\left( m_{1}+...+m_{n-1}\right) \right\rceil ,\text{ }m_{n-1}\right\} \right\} -1$ when $n\geq 2$ and $\gamma _{g}^{\prime }(L\left( K_{\overline{m}}\right) )=\min \left\{ \left\lceil \frac{2}{3}\left( \left\vert V(K_{_{\overline{m}}})\right\vert -2\right) \right\rceil ,\right.$ $\left. 2\max \left\{ \left\lceil \frac{1}{2}\left( m_{1}+...+m_{n-1}-1\right) \right\rceil ,\text{ }m_{n-1}\right\} \right\}$ when $n\geq 4$.Comment: 7 page

On the game total domination number

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 $G$, $\gamma_{{\rm tg}}(G)$, is the number of vertices chosen when Dominator starts the game and both players play optimally. Recently, Henning, Klav\v{z}ar, and Rall proved that $\gamma_{{\rm tg}}(G) \le \frac{4}{5}n$ holds for every graph $G$ which is given on $n$ vertices such that every component of it is of order at least $3$; they also conjectured that the sharp upper bound would be $\frac{3}{4}n$. Here, we prove that $\gamma_{{\rm tg}}(G)\le \frac{11}{14}n$ holds for every $G$ which contains no isolated vertices or isolated edges.Comment: 11 page

Zero forcing number, Grundy domination number, and their variants

This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem. We show that the Grundy domination type parameters are bounded above by the minimum rank type parameters. We also give a method to calculate the $L$-Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the $L$-Grundy domination number

Complexity of the Game Domination Problem

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.Comment: 14 pages, 3 figure

Cutting Lemma and Union Lemma for the Domination Game

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 listed

Dominating sequences under atomic changes with applications in Sierpi\'{n}ski and interval graphs

A sequence $S=(v_1,\ldots,v_k)$ of distinct vertices of a graph $G$ is called a legal sequence if $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j]\not=\emptyset$ for any $i$. The maximum length of a legal (dominating) sequence in $G$ is called the Grundy domination number $\gamma_{gr}(G)$ of a graph $G$. It is known that the problem of determining the Grundy domination number is NP-complete in general, while efficient algorithm exist for trees and some other classes of graphs. In this paper we find an efficient algorithm for the Grundy domination number of an interval graph. We also show the exact value of the Grundy domination number of an arbitrary Sierpi\'{n}ski graph $S_p^n$, and present algorithms to construct the corresponding sequence. These results are obtained by using the main result of the paper, which are sharp bounds for the Grundy domination number of a vertex- and edge-removed graph. That is, given a graph $G$, $e\in E(G)$, and $u\in V(G)$, we prove that $\gamma_{gr}(G)-1\le \gamma_{gr}(G-e) \le \gamma_{gr}(G)+1$ and $\gamma_{gr}(G)-2\le \gamma_{gr}(G-u) \le \gamma_{gr}(G)$. For each of the bounds there exist graphs, in which all three possibilities occur for different edges, respectively vertices.Comment: 15 pages, 4 figure

Progress Towards the Total Domination Game 34\frac{3}{4}-Conjecture

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, $\gamma_{\rm tg}(G)$, of $G$ is the number of vertices chosen when Dominator starts the game and both players play optimally. Henning, Klav\v{z}ar and Rall [Combinatorica, to appear] posted the $\frac{3}{4}$-Game Total Domination Conjecture that states that if $G$ is a graph on $n$ vertices in which every component contains at least three vertices, then $\gamma_{\rm tg}(G) \le \frac{3}{4}n$. In this paper, we prove this conjecture over the class of graphs $G$ that satisfy both the condition that the degree sum of adjacent vertices in $G$ is at least $4$ and the condition that no two vertices of degree $1$ are at distance $4$ apart in $G$. In particular, we prove that by adopting a greedy strategy, Dominator can complete the total domination game played in a graph with minimum degree at least $2$ in at most $3n/4$ moves.Comment: 14 page

Domination game and minimal edge cuts

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 $\gamma_g(G) \le \gamma_g(G\setminus C) + 2\kappa'(G)$. 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. Along the way two technical lemmas, which seem to be generally applicable for the study of the domination game, are proved.Comment: 15 pages, 1 figur