scholarly journals Total and Double Total Domination Number on Hexagonal Grid

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1110
Author(s):  
Antoaneta Klobučar ◽  
Ana Klobučar
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Hexagonal Grid

In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid H m , n with m hexagons in a row and n hexagons in a column. Further, we explore the ratio between the total domination number and the number of vertices of H m , n when m and n tend to infinity.

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.

scholarly journals Locating-Total Domination Number of Cacti Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jianxin Wei ◽  
Uzma Ahmad ◽  
Saira Hameed ◽  
Javaria Hanif
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Unicyclic Graphs ◽  
Total Dominating Set ◽  
Bicyclic Graphs ◽  
Number Of Cycles

For a connected graph J, a subset W ⊆ V J is termed as a locating-total dominating set if for a ∈ V J ,   N a ∩ W ≠ ϕ , and for a ,   b ∈ V J − W ,   N a ∩ W ≠ N b ∩ W . The number of elements in a smallest such subset is termed as the locating-total domination number of J. In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.

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 $n/2$.

scholarly journals Some results on matching and total domination in graphs

2010 ◽  
Vol 4 (2) ◽  
pp. 241-252 ◽  
Author(s):  
Chee Shiu ◽  
Xue-Gang Chen ◽  
Hong Chan
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Maximum Matching ◽  
Matching Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Total Dominating Set ◽  
Domination In Graphs

Let G be a graph. A set S of vertices of G is called a total dominating set of G if every vertex of G is adjacent to at least one vertex in S. The total domination number ?t(G) and the matching number ??(G) of G are the cardinalities of the minimum total dominating set and the maximum matching of G, respectively. In this paper, we will introduce an upper bound of the di?erence between ?t(G) and ??(G). We will also characterize every tree T with ?t(T)? ??(T), and give a family of graphs with ?t(G)???(G).

scholarly journals Nordhaus-Gaddum Type Results for Total Domination

2011 ◽  
Vol Vol. 13 no. 3 (Graph Theory) ◽  
Author(s):  
Michael Henning ◽  
Ernst Joubert ◽  
Justin Southey
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Domination Number ◽  
Total Domination ◽  
Type Result ◽  
Total Domination Number ◽  
Maximum Value ◽  
International Audience ◽  
Domination Numbers

Graph Theory International audience A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s,s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of G1 and G2 is 2s+4, with equality if and only if G1 = sK2 or G2 = sK2, while the maximum value of the product of the total domination numbers of G1 and G2 is max{8s,⌊(s+6)2/4 ⌋}.

scholarly journals The 4/5 upper bound on the game total domination number

COMBINATORICA ◽  
2016 ◽  
Vol 37 (2) ◽  
pp. 223-251 ◽  
Author(s):  
Michael A. Henning ◽  
Sandi Klavžar ◽  
Douglas F. Rall
Keyword(s):  
Upper Bound ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number

A new lower bound on the total domination number of a graph

2022 ◽  
pp. 1-14
Author(s):  
Majid Hajian ◽  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Lower Bound ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number
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