scholarly journals The Graphs Whose Sum of Global Connected Domination Number and Chromatic Number is 2n-5

2012 ◽  
Vol 11 (4) ◽  
pp. 91-98 ◽  
Author(s):  
Mahadevan G ◽  
A Selvam Avadayappan ◽  
Twinkle Johns
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Extremal Graphs ◽  
Minimum Cardinality ◽  
Connected Domination Number

A subset S of vertices in a graph G = (V,E) is a dominating set if every vertex in V-S is adjacent to atleast one vertex in S. A dominating set S of a connected graph G is called a connected dominating set if the induced sub graph < S > is connected. A set S is called a global dominating set of G if S is a dominating set of both G and . A subset S of vertices of a graph G is called a global connected dominating set if S is both a global dominating and a connected dominating set. The global connected domination number is the minimum cardinality of a global connected dominating set of G and is denoted by γgc(G). In this paper we characterize the classes of graphs for which γgc(G) + χ(G) = 2n-5 and 2n-6 of global connected domination number and chromatic number and characterize the corresponding extremal graphs.

scholarly journals On the doubly connected domination number of a graph

2006 ◽  
Vol 4 (1) ◽  
pp. 34-45 ◽  
Author(s):  
Joanna Cyman ◽  
Magdalena Lemańska ◽  
Joanna Raczek
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Connected Domination Number

AbstractFor a given connected graph G = (V, E), a set $$D \subseteq V(G)$$ is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.

scholarly journals Inverse Connected and Disjoint Connected Domination Number of a Jump Graph

2018 ◽  
Vol 7 (4.10) ◽  
pp. 585 ◽  
Author(s):  
Annie Jasmine.S.E ◽  
K. Ameenal Bibi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Minimum Cardinality ◽  
Graph Theoretic ◽  
Minimum Connected Dominating Set ◽  
Connected Domination Number

Let D be the minimum connected dominating set of a jump graph . If  of  contains a connected dominating set , then  is called the inverse connected dominating set of the jump graph . The minimum cardinality of an inverse connected dominating set is the inverse connected domination number of the jump graph, denoted by. The disjoint connected domination number, of the jump graph , is the minimum cardinality of the union of two disjoint connected dominating set of  . In this paper we have established bounds, exact values of and graph theoretic relations between the inverse connected domination number of the jump graph with other parameters of G.       

scholarly journals Blast Domination Number of Transformation Graphs of Linear and Circular Graphs

2020 ◽  
Vol 8 (4S4) ◽  
pp. 93-96
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Domination Numbers

A subset S of V of a non-trivial connected graph G is called a Blast dominating set (BD-set), if S is a connected dominating set and the induced sub graph 𝑽 − 𝑺 is triple connected. The minimum cardinality taken over all such Blast Dominating sets is called the Blast Domination Number (BDN) of G and is denoted as, 𝜸𝒄 𝒕𝒄 (𝑮). In this article, let us mull over the generalized transformation graphs 𝑮 𝒂𝒃 and get hold of the analogous lexis of the Blast domination numbers for all the rage, transformation graphs, 𝑮 𝒂𝒃 and their complement graphs, 𝑮 𝒂 𝒃 for linear and circular graphs.

scholarly journals Nordhaus$-$Gaddum type results for connected and total domination

2020 ◽  
Author(s):  
Rana Khoeilar ◽  
Hossein Karami ◽  
Mustapha Chellali ◽  
Seyed Mahmoud Sheikholeslami ◽  
Lutz Volkmann
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Vertex Degree ◽  
Total Domination ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Connected Domination Number ◽  
Delta G

A dominating set of $G=(V,E)$ is a subset $S$ of $V$ such that every vertex in $V-S$ has at least one neighbor in $S.$ A connected dominating set of $G$ is a dominating set whose induced subgraph is connected. The minimum cardinality of a connected dominating set is the connected domination number $\gamma _{c}(G)$. Let $\delta ^{\ast }(G)=\min \{\delta (G),\delta (% \overline{G})\}$, where $\overline{G}$ is the complement of $G$ and $\delta (G)$ is the minimum vertex degree. In this paper, we improve upon existing results by providing new Nordhaus-Gaddum type results for connected domination. In particular, we show that if $G$ and $\overline{G}$ are both connected and $\min \{\gamma _{c}(G),\gamma _{c}(\overline{G})\}\geq 3$, then $\gamma _{c}(G)+\gamma _{c}(\overline{G})\leq 4+(\delta ^{\ast }(G)-1)(% \frac{1}{\gamma _{c}(G)-2}+\frac{1}{\gamma _{c}(\overline{G})-2})$ and $% \gamma _{c}(G)\gamma _{c}(\overline{G})\leq 2(\delta ^{\ast }(G)-1)(\frac{1}{% \gamma _{c}(G)-2}+\frac{1}{\gamma _{c}(\overline{G})-2}+\frac{1}{2})+4$. Moreover, we will establish accordingly results for total domination.

TOTAL OUTER-CONNECTED DOMINATION SUBDIVISION NUMBERS IN GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350009
Author(s):  
O. FAVARON ◽  
R. KHOEILAR ◽  
S. M. SHEIKHOLESLAMI
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Connected Dominating Set ◽  
Minimum Size ◽  
Connected Domination Number ◽  
Minimum Number ◽  
Domination Subdivision Number

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph G[V\S] induced by V\S is connected. The total outer-connected domination numberγ toc (G) is the minimum size of such a set. The total outer-connected domination subdivision number sd γ toc (G) is the minimum number of edges that must be subdivided in order to increase the total outer-connected domination number. We prove the existence of sd γ toc (G) for every connected graph G of order at least 3 and give upper bounds on it in some classes of graphs.

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

The minimum eccentric distance sum of trees with given distance k-domination number

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
Author(s):  
Lidan Pei ◽  
Xiangfeng Pan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Simple Connected Graph ◽  
Eccentric Distance ◽  
Distance Formula

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

On the minimum vertex covering transversal dominating sets in graphs and their classification

2017 ◽  
Vol 09 (05) ◽  
pp. 1750069 ◽  
Author(s):  
R. Vasanthi ◽  
K. Subramanian
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Covering ◽  
Number Formula ◽  
Covering Set ◽  
Simple Connected Graph

Let [Formula: see text] be a simple and connected graph. A dominating set [Formula: see text] is said to be a vertex covering transversal dominating set if it intersects every minimum vertex covering set of [Formula: see text]. The vertex covering transversal domination number [Formula: see text] is the minimum cardinality among all vertex covering transversal dominating sets of [Formula: see text]. A vertex covering transversal dominating set of minimum cardinality [Formula: see text] is called a minimum vertex covering transversal dominating set or simply a [Formula: see text]-set. In this paper, we prove some general theorems on the vertex covering transversal domination number of a simple connected graph. We also provide some results about [Formula: see text]-sets and try to classify those sets based on their intersection with the minimum vertex covering sets.

scholarly journals Global Alliances and Independent Domination in Some Classes of Graphs

10.37236/847 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Odile Favaron
Keyword(s):  
Bipartite Graph ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Extremal Graphs ◽  
Minimum Cardinality ◽  
Independent Domination

A dominating set $S$ of a graph $G$ is a global (strong) defensive alliance if for every vertex $v\in S$, the number of neighbors $v$ has in $S$ plus one is at least (greater than) the number of neighbors it has in $V\setminus S$. The dominating set $S$ is a global (strong) offensive alliance if for every vertex $v\in V\setminus S$, the number of neighbors $v$ has in $S$ is at least (greater than) the number of neighbors it has in $V\setminus S$ plus one. The minimum cardinality of a global defensive (strong defensive, offensive, strong offensive) alliance is denoted by $\gamma_a(G)$ ($\gamma_{\hat a}(G)$, $\gamma_o(G)$, $\gamma_{\hat o}(G))$. We compare each of the four parameters $\gamma_a, \gamma_{\hat a}, \gamma_o, \gamma_{\hat o}$ to the independent domination number $i$. We show that $i(G)\le \gamma ^2_a(G)-\gamma_a(G)+1$ and $i(G)\le \gamma_{\hat{a}}^2(G)-2\gamma_{\hat{a}}(G)+2$ for every graph; $i(G)\le \gamma ^2_a(G)/4 +\gamma_a(G)$ and $i(G)\le \gamma_{\hat{a}}^2(G)/4 +\gamma_{\hat{a}}(G)/2$ for every bipartite graph; $i(G)\le 2\gamma_a(G)-1$ and $i(G)=3\gamma_{\hat{a}}(G)/2 -1$ for every tree and describe the extremal graphs; and that $\gamma_o(T)\le 2i(T)-1$ and $i(T)\le \gamma_{\hat o}(T)-1$ for every tree. We use a lemma stating that $\beta(T)+2i(T)\ge n+1$ in every tree $T$ of order $n$ and independence number $\beta(T)$.

scholarly journals Co-Secure Set Domination in Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 66-68
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Secure Set ◽  
Domination In Graphs

Throughout this paper, consider G = (V,E) as a connected graph. A subset D of V(G) is a set dominating set of G if for every M  V / D there exists a non-empty set N of D such that the induced sub graph <MUN> is connected. A subset D of the vertex set of a graph G is called a co-secure dominating set of a graph if D is a dominating set, and for each u' D there exists a vertex v'V / D such that u'v' is an edge and D \u'v' is a dominating set. A co-secure dominating set D is a co-secure set dominating set of G if D is also a set dominating set of G. The co-secure set domination number G s cs γ is the minimum cardinality of a co-secure set dominating set. In this paper we initiate the study of this new parameter & also determine the co-secure set domination number of some standard graphs and obtain its bounds.

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