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.

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 Total connected domination game

2021 ◽  
Vol 41 (4) ◽  
pp. 453-464
Author(s):  
Csilla Bujtás ◽  
Michael A. Henning ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Large Family ◽  
Infinite Family ◽  
Connected Subgraph ◽  
Product Graphs ◽  
Class 1 ◽  
Game Domination Number ◽  
Total Domination Game ◽  
Domination Game

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.

scholarly journals Competition-Independence Game and Domination Game

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 359 ◽  
Author(s):  
Chalermpong Worawannotai ◽  
Watcharintorn Ruksasakchai
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Optimal Size ◽  
Independence Numbers ◽  
Game Domination Number ◽  
Domination Game ◽  
Domination Numbers

The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as many turns as possible. The game ends when all vertices are dominated. The game domination number, denoted by γ g ( G ) (respectively γ g ′ ( G ) ), is the total number of turns when both players play optimally and when Dominator (respectively Staller) starts the game. In this paper, we study a version of this game where the set of chosen vertices is always independent. This version turns out to be another game known as the competition-independence game. The competition-independence game is played on a graph by two players, Diminisher and Sweller. They take turns in constructing maximal independent set M, where Diminisher tries to minimize | M | and Sweller tries to maximize | M | . Note that, actually, it is the domination game in which the set of played vertices is independent. The competition-independence number, denoted by I d ( G ) (respectively I s ( G ) ) is the optimal size of the final independent set in the competition-independence game if Diminisher (respectively Sweller) starts the game. In this paper, we check whether some well-known results in the domination game hold for the competition-independence game. We compare the competition-independence numbers to the game domination numbers. Moreover, we provide a family of graphs such that many parameters are equal. Finally, we present a realization result on the competition-independence numbers.

scholarly journals On the Game Domination Number of Graphs with Given Minimum Degree

10.37236/4497 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Csilla Bujtás
Keyword(s):  
Upper Bound ◽  
Minimum Degree ◽  
Domination Number ◽  
Game Domination Number ◽  
Domination Game

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 $\gamma_g(G)$ of $G$ is the length of the domination game in which Dominator starts and both players play optimally. In this paper we establish an upper bound on $\gamma_g(G)$ in terms of the minimum degree $\delta$ and the order $n$ of $G$. Our main result states that for every $\delta \ge 4$,$$\gamma_g(G)\le \frac{15\delta^4-28\delta^3-129\delta^2+354\delta-216}{45\delta^4-195\delta^3+174\delta^2+174\delta-216}\; n.$$Particularly, $\gamma_g(G) < 0.5139\; n$ holds for every graph of minimum degree 4, and $\gamma_g(G) < 0.4803\; n$ if the minimum degree is greater than 4. Additionally, we prove that $\gamma_g(G) < 0.5574\; n$ if $\delta=3$.

scholarly journals Connected domination game

2019 ◽  
Vol 13 (1) ◽  
pp. 261-289 ◽  
Author(s):  
Mieczysław Borowiecki ◽  
Anna Fiedorowicz ◽  
Elżbieta Sidorowicz
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Minimum Cardinality ◽  
Legal Move ◽  
Game Domination Number ◽  
Domination Game

In this paper we introduce a domination game based on the notion of connected domination. Let G = (V,E) be a connected graph of order at least 2. We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller. The players alternate taking turns choosing a vertex of G (Dominator starts). A move of a player by choosing a vertex v is legal, if (1) the vertex v dominates at least one additional vertex that was not dominated by the set of previously chosen vertices and (2) the set of all chosen vertices induces a connected subgraph of G. The game ends when none of the players has a legal move (i.e., G is dominated). The aim of Dominator is to finish as soon as possible, Staller has an opposite aim. Let D be the set of played vertices obtained at the end of the connected domination game (D is a connected dominating set of G). The connected game domination number of G, denoted cg(G), is the minimum cardinality of D, when both players played optimally on G. We provide an upper bound on cg(G) in terms of the connected domination number. We also give a tight upper bound on this parameter for the class of 2-trees. Next, we investigate the Cartesian product of a complete graph and a tree, and we give exact values of the connected game domination number for such a product, when the tree is a path or a star. We also consider some variants of the game, in particular, a Staller-start game.

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

LOGARITHMIC-TIME LINK PATH QUERIES IN A SIMPLE POLYGON

1995 ◽  
Vol 05 (04) ◽  
pp. 369-395 ◽  
Author(s):  
ESTHER M. ARKIN ◽  
JOSEPH S.B. MITCHELL ◽  
SUBHASH SURI
Keyword(s):  
Data Structure ◽  
Shortest Path ◽  
Simple Polygon ◽  
Additive Constant ◽  
Time And Space ◽  
Link Distance ◽  
Path Distance ◽  
Path Queries ◽  
Shortest Path Distance ◽  
Small Additive

We develop a data structure for answering link distance queries between two arbitrary points in a simple polygon. The data structure requires O(n3) time and space for its construction and answers link distance queries in O(log n) time, after which a minimum-link path can be reported in time proportional to the number of links. Here, n denotes the number of vertices of the polygon. Our result extends to link distance queries between pairs of segments or polygons. We also propose a simpler data structure for computing a link distance approximately, where the error is bounded by a small additive constant. Finally, we also present a scheme for approximating the link and the shortest path distance simultaneously.

Sign in / Sign up

Export Citation Format

Share Document