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.

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.

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.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

Bounds for complete cototal domination number of Cartesian product graphs and complement graphs

2021 ◽  
pp. 2150090
Author(s):  
J. Maria Regila Baby ◽  
K. Uma Samundesvari
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Product Graphs ◽  
Cartesian Product Graphs

A total dominating set [Formula: see text] is said to be a complete cototal dominating set if [Formula: see text] has no isolated nodes and it is represented by [Formula: see text]. The complete cototal domination number, represented by [Formula: see text], is the minimum cardinality of a [Formula: see text] set of [Formula: see text]. In this paper, the bounds for complete cototal domination number of Cartesian product graphs and complement graphs are determined.

Total edge–vertex domination

2020 ◽  
Vol 54 ◽  
pp. 1 ◽  
Author(s):  
Abdulgani Sahin ◽  
Bünyamin Sahin
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Np Hard ◽  
Minimum Cardinality

An edge e ev-dominates a vertex v which is a vertex of e, as well as every vertex adjacent to v. A subset D ⊆ E is an edge-vertex dominating set (in simply, ev-dominating set) of G, if every vertex of a graph G is ev-dominated by at least one edge of D. The minimum cardinality of an ev-dominating set is named with ev-domination number and denoted by γev(G). A subset D ⊆ E is a total edge-vertex dominating set (in simply, total ev-dominating set) of G, if D is an ev-dominating set and every edge of D shares an endpoint with other edge of D. The total ev-domination number of a graph G is denoted with γevt(G) and it is equal to the minimum cardinality of a total ev-dominating set. In this paper, we initiate to study total edge-vertex domination. We first show that calculating the number γevt(G) for bipartite graphs is NP-hard. We also show the upper bound γevt(T) ≤ (n − l + 2s − 1)∕2 for the total ev-domination number of a tree T, where T has order n, l leaves and s support vertices and we characterize the trees achieving this upper bound. Finally, we obtain total ev-domination number of paths and cycles.

scholarly journals A New Upper Bound on the Total Domination Number of a Graph

10.37236/983 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than $2$ and we let ${\cal L}$ be the set of all large vertices of $G$. Let $P$ be any component of $G - {\cal L}$; it is a path. If $|P| \equiv 0 \, ( {\rm mod} \, 4)$ and either the two ends of $P$ are adjacent in $G$ to the same large vertex or the two ends of $P$ are adjacent to different, but adjacent, large vertices in $G$, we call $P$ a $0$-path. If $|P| \ge 5$ and $|P| \equiv 1 \, ( {\rm mod} \, 4)$ with the two ends of $P$ adjacent in $G$ to the same large vertex, we call $P$ a $1$-path. If $|P| \equiv 3 \, ( {\rm mod} \, 4)$, we call $P$ a $3$-path. For $i \in \{0,1,3\}$, we denote the number of $i$-paths in $G$ by $p_i$. We show that the total domination number of $G$ is at most $(n + p_0 + p_1 + p_3)/2$. This result generalizes a result shown in several manuscripts (see, for example, J. Graph Theory 46 (2004), 207–210) which states that if $G$ is a graph of order $n$ with minimum degree at least three, then the total domination of $G$ is at most $n/2$. It also generalizes a result by Lam and Wei stating that if $G$ is a graph of order $n$ with minimum degree at least two and with no degree-$2$ vertex adjacent to two other degree-$2$ vertices, then the total domination of $G$ is at most $n/2$.

Bounds on 2-point set domination number of a graph

2018 ◽  
Vol 10 (01) ◽  
pp. 1850012
Author(s):  
Purnima Gupta ◽  
Deepti Jain
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Extremal Graphs ◽  
Minimum Cardinality ◽  
Point Set

A set [Formula: see text] is a [Formula: see text]-point set dominating set (2-psd set) of a graph [Formula: see text] if for any subset [Formula: see text], there exists a nonempty subset [Formula: see text] containing at most two vertices such that the subgraph [Formula: see text] induced by [Formula: see text] is connected. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. In this paper, we determine the lower bounds and an upper bound on [Formula: see text] of a graph. We also characterize extremal graphs for the lower bounds and identify some well-known classes of both separable and nonseparable graphs attaining the upper bound.

The k-power bondage number of a graph

2016 ◽  
Vol 08 (04) ◽  
pp. 1650064
Author(s):  
Seethu Varghese ◽  
A. Vijayakumar
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Power Domination ◽  
Bondage Number ◽  
Number Formula

The [Formula: see text]-power domination number, [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of a [Formula: see text]-power dominating set of [Formula: see text]. In this paper, we initiate the study of the [Formula: see text]-power bondage number, [Formula: see text], of a graph [Formula: see text], i.e., the minimum cardinality among all sets [Formula: see text] for which [Formula: see text]. We obtain a sharp upper bound for [Formula: see text] in terms of the degree of [Formula: see text]. We prove that [Formula: see text] for any nonempty tree [Formula: see text] and also provide some conditions on [Formula: see text] for [Formula: see text].

scholarly journals Properties of the Global Total k-Domination Number

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 480
Author(s):  
Frank A. Hernández Mira ◽  
Ernesto Parra Inza ◽  
José M. Sigarreta Almira ◽  
Nodari Vakhania
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Minimum Cardinality

A nonempty subset D⊂V of vertices of a graph G=(V,E) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D⊆V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v∈V, and it is a global total k-dominating set if D is a total k-dominating set of both G and G¯. The global total k-domination number of G, denoted by γktg(G), is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of γktg(G), and develop a method that generates a GTkD-set from a GT(k−1)D-set for the successively increasing values of k. Based on this method, we establish a relationship between γ(k−1)tg(G) and γktg(G), which, in turn, provides another upper bound on γktg(G).

scholarly journals On autographix conjecture regarding domination number and average eccentricity

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 699-710 ◽  
Author(s):  
Li-Dan Pei ◽  
Xiang-Feng Pan ◽  
Jing Tian ◽  
Gui-Qin Peng
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Mean Value ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Maximal Distance ◽  
The Mean ◽  
Average Eccentricity

The eccentricity of a vertex is the maximal distance from it to another vertex and the average eccentricity ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. A set S ? V(G) is a dominating set of a graph G if NG(v) ? S ? 0 for any vertex v ? V(G)\S. The domination number (G) of G is the minimum cardinality of all dominating sets of G. In this paper, we correct an AutoGraphiX conjecture regarding the domination number and average eccentricity, and present a proof of the revised conjecture. In addition, we establish an upper bound on ?(T)-ecc(T) for an n-vertex tree T.

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