scholarly journals On Two Open Problems on Double Vertex-Edge Domination in Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1010
Author(s):  
Fang Miao ◽  
Wenjie Fan ◽  
Mustapha Chellali ◽  
Rana Khoeilar ◽  
Seyed Mahmoud Sheikholeslami ◽  
...  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Open Problems ◽  
Free Graph ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Edge Dominating Set ◽  
Domination In Graphs

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G. A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V - S has at least two neighbors in S). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G, and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with γ d v e ( T ) = γ t ( T ) or γ d v e ( T ) = γ 2 ( T ) .

scholarly journals Complete Cototal Domination Number of a Graph

2011 ◽  
Vol 3 (3) ◽  
pp. 547-555 ◽  
Author(s):  
B. Basavanagoud ◽  
S. M. Hosamani
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Open Problems ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination In Graphs

Let  be a graph. A set  of a graph  is called a total dominating set if the induced subgraph  has no isolated vertices. The total domination number  of G is the minimum cardinality of a total dominating set of G. A total dominating set D is said to be a complete cototal dominating set if the induced subgraph  has no isolated vertices. The complete cototal domination number  of G is the minimum cardinality of a complete cototal dominating set of G. In this paper, we initiate the study of complete cototal domination in graphs and present bounds and some exact values for . Also its relationship with other domination parameters are established and related two open problems are explored.Keywords: Domination number; Total domination number; Cototal domination number; Complete cototal domination number.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i3.7744               J. Sci. Res. 3 (3), 557-565 (2011)

scholarly journals Total Domination and Matching Numbers in Claw-Free Graphs

10.37236/1085 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Matching Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Free Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $M$ of edges of a graph $G$ is a matching if no two edges in $M$ are incident to the same vertex. The matching number of $G$ is the maximum cardinality of a matching of $G$. A set $S$ of vertices in $G$ 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 $G$. If $G$ does not contain $K_{1,3}$ as an induced subgraph, then $G$ is said to be claw-free. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number. In this paper, we use transversals in hypergraphs to characterize connected claw-free graphs with minimum degree at least three that have equal total domination and matching numbers.

scholarly journals On k-Fair Total Domination in Graphs

2021 ◽  
Vol 14 (2) ◽  
pp. 578-589
Author(s):  
Wardah Masanggila Bent-Usman ◽  
Rowena T. Isla
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination In Graphs ◽  
Product Of Graphs

Let G = (V (G), E(G)) be a simple non-empty graph. For an integer k ≥ 1, a k-fairtotal dominating set (kf td-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkf td(G), is the minimum cardinality of a kf td-set. A k-fair total dominating set of cardinality γkf td(G) is called a minimum k-fair total dominating set or a γkf td-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharpbounds of their corresponding k-fair total domination number.

scholarly journals Locating-Total Dominating Sets in Twin-Free Graphs: a Conjecture

10.37236/5147 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Florent Foucaud ◽  
Michael A. Henning
Keyword(s):  
Dominating Set ◽  
Open Neighborhood ◽  
Domination Number ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Free Graph ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods.  The location-total domination number of $G$, denoted $\gamma_t^L(G)$, is the minimum cardinality of a locating-total dominating set in $G$. It is well-known that every connected graph of order $n \ge 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twin-free graph of order $n$ with no isolated vertex, then $\gamma_t^L(G) \le \frac{2}{3}n$. We prove the conjecture for graphs without $4$-cycles as a subgraph. We also prove that if $G$ is a twin-free graph of order $n$, then $\gamma_t^L(G) \le \frac{3}{4}n$.

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 New Bounds on the Double Total Domination Number of Graphs

2021 ◽  
Author(s):  
A. Cabrera-Martínez ◽  
F. A. Hernández-Mira
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

AbstractLet G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

scholarly journals ON THE ROOTS OF TOTAL DOMINATION POLYNOMIAL OF GRAPHS, II

2019 ◽  
pp. 659
Author(s):  
Saeid Alikhani ◽  
Nasrin Jafari
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination Polynomial

Let $G = (V, E)$ be a simple graph of order $n$. A  total dominating set of $G$ is a subset $D$ of $V$, such that every vertex of $V$ is adjacent to at least one vertex in  $D$. The total domination number of $G$ is  minimum cardinality of  total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of the total domination polynomial of some graphs.  We show that  all roots of $D_t(G, x)$ lie in the circle with center $(-1, 0)$ and radius $\sqrt[\delta]{2^n-1}$, where $\delta$ is the minimum degree of $G$. As a consequence, we prove that if $\delta\geq \frac{2n}{3}$,  then every integer root of $D_t(G, x)$ lies in the set $\{-3,-2,-1,0\}$.

scholarly journals Minimum Total Dominating Energy of Some Standard Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 272-276
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Ladder Graph ◽  
Eigen Values ◽  
Vertex Set

Let G be a simple graph with vertex set V(G) and edge set E(G). A set S of vertices in a graph 𝑮(𝑽,𝑬) is called a total dominating set if every vertex 𝒗 ∈ 𝑽 is adjacent to an element of S. The minimum cardinality of a total dominating set of G is called the total domination number of G which is denoted by 𝜸𝒕 (𝑮). The energy of the graph is defined as the sum of the absolute values of the eigen values of the adjacency matrix. In this paper, we computed minimum total dominating energy of a Friendship Graph, Ladder Graph and Helm graph. The Minimum total dominating energy for bistar graphand sun graph is 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).

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