scholarly journals Extremal Problems for Game Domination Number

2013 ◽  
Vol 27 (4) ◽  
pp. 2090-2107 ◽  
Author(s):  
William B. Kinnersley ◽  
Douglas B. West ◽  
Reza Zamani
Keyword(s):  
Domination Number ◽  
Extremal Problems ◽  
Game Domination Number

On graphs with largest possible game domination number

2018 ◽  
Vol 341 (6) ◽  
pp. 1768-1777 ◽  
Author(s):  
Kexiang Xu ◽  
Xia Li ◽  
Sandi Klavžar
Keyword(s):  
Domination Number ◽  
Game Domination Number

scholarly journals Guarded subgraphs and the domination game

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Boštjan Brešar ◽  
Sandi Klavžar ◽  
Gasper Košmrlj ◽  
Doug F. Rall
Keyword(s):  
Graph Theory ◽  
Domination Number ◽  
Metric Properties ◽  
International Audience ◽  
Game Domination Number ◽  
Domination Game

Graph Theory International audience We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.

scholarly journals On graphs with small game domination number

2016 ◽  
Vol 10 (1) ◽  
pp. 30-45 ◽  
Author(s):  
Sandi Klavzar ◽  
Gasper Kosmrlj ◽  
Simon Schmidt
Keyword(s):  
Domination Number ◽  
Small Game ◽  
Game Domination Number ◽  
Domination Game

The domination game is played on a graph G by Dominator and Staller. The game domination number ?(G) of G is the number of moves played when Dominator starts and both players play optimally. Similarly, ?g (G) is the number of moves played when Staller starts. Graphs G with ?(G) = 2, graphs with ?g(G) = 2, as well as graphs extremal with respect to the diameter among these graphs are characterized. In particular, ?g (G) = 2 and diam(G) = 3 hold for a graph G if and only if G is a so-called gamburger. Graphs G with ?(G) = 3 and diam(G) = 6, as well as graphs G with ?g(G) = 3 and diam(G) = 5 are also characterized.

scholarly journals Connected domination game played on Cartesian products

2019 ◽  
Vol 17 (1) ◽  
pp. 1269-1280 ◽  
Author(s):  
Csilla Bujtás ◽  
Pakanun Dokyeesun ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Lower Bound ◽  
Domination Number ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Arbitrary Graph ◽  
Cartesian Products ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Game Domination Number ◽  
Domination Game

Abstract The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard 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 connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

THE DOMINATION GAME ON SPLIT GRAPHS

2018 ◽  
Vol 99 (2) ◽  
pp. 327-337 ◽  
Author(s):  
TIJO JAMES ◽  
SANDI KLAVŽAR ◽  
AMBAT VIJAYAKUMAR
Keyword(s):  
Domination Number ◽  
Split Graph ◽  
General Bound ◽  
Even Order ◽  
Split Graphs ◽  
Game Domination Number ◽  
Domination Game

We investigate the domination game and the game domination number $\unicode[STIX]{x1D6FE}_{g}$ in the class of split graphs. We prove that $\unicode[STIX]{x1D6FE}_{g}(G)\leq n/2$ for any isolate-free $n$-vertex split graph $G$, thus strengthening the conjectured $3n/5$ general bound and supporting Rall’s $\lceil n/2\rceil$-conjecture. We also characterise split graphs of even order with $\unicode[STIX]{x1D6FE}_{g}(G)=n/2$.

scholarly journals Fractional Domination Game

10.37236/8730 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Csilla Bujtas ◽  
Zsolt Tuza
Keyword(s):  
Domination Number ◽  
Fundamental Property ◽  
Discrete Math ◽  
Additive Constant ◽  
Graph Domination ◽  
Continuation Principle ◽  
Fractional Domination ◽  
Game Domination Number ◽  
Domination Game ◽  
Small Additive

Given a graph $G$, a real-valued function $f: V(G) \rightarrow [0,1]$ is a fractional dominating function if $\sum_{u \in N[v]} f(u) \ge 1$ holds for every vertex $v$ and its closed neighborhood $N[v]$ in $G$. The aim is to minimize the sum $\sum_{v \in V(G)} f(v)$. A different approach to graph domination is the domination game, introduced by Brešar et al. [SIAM J. Discrete Math. 24 (2010) 979–991]. It is played on a graph $G$ by two players, namely Dominator and Staller, who take turns choosing a vertex such that at least one previously undominated vertex becomes dominated. The game is over when all vertices are dominated. Dominator wants to finish the game as soon as possible, while Staller wants to delay the end. Assuming that both players play optimally and Dominator starts, the length of the game on $G$ is uniquely determined and is called the game domination number of $G$. We introduce and study the fractional version of the domination game, where the moves are ruled by the condition of fractional domination. Here we prove a fundamental property of this new game, namely the fractional version of the so-called Continuation Principle. Moreover, we present lower and upper bounds on the fractional game domination number of paths and cycles. These estimates are tight apart from a small additive constant. We also prove that the game domination number cannot be bounded above by any linear function of the fractional game domination number.

Roman game domination number of a graph

2016 ◽  
Vol 33 (2) ◽  
pp. 713-725 ◽  
Author(s):  
A. Bahremandpour ◽  
S. M. Sheikholeslami ◽  
L. Volkmann
Keyword(s):  
Domination Number ◽  
Game Domination Number
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