A SIMPLE HEURISTIC FOR MINIMUM CONNECTED DOMINATING SET IN GRAPHS

2003 ◽  
Vol 14 (02) ◽  
pp. 323-333 ◽  
Author(s):  
PENG-JUN WAN ◽  
KHALED M. ALZOUBI ◽  
OPHIR FRIEDER
Keyword(s):  
Planar Graph ◽  
Complete Graph ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Minimum Connected Dominating Set ◽  
Simple Heuristic ◽  
Connected Domination Number ◽  
A Minor

Let α2(G), γ(G) and γc(G) be the 2-independence number, the domination number, and the connected domination number of a graph G respectively. Then α2(G) ≤ γ (G) ≤ γc(G). In this paper , we present a simple heuristic for Minimum Connected Dominating Set in graphs. When running on a graph G excluding Km (the complete graph of order m) as a minor, the heuristic produces a connected dominating set of cardinality at most 7α2(G) - 4 if m = 3, or at most [Formula: see text] if m ≥ 4. In particular, if running on a planar graph G, the heuristic outputs a connected dominating set of cardinality at most 15α2(G) - 5.

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.       

Clique doubly connected domination in the join and lexicographic product of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050066
Author(s):  
Enrico L. Enriquez ◽  
Albert D. Ngujo
Keyword(s):  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Connected Dominating Set ◽  
Lexicographic Product ◽  
Connected Domination Number ◽  
Vertex Set ◽  
Product Of Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two 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 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 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.

Outer-connected domination in 2-connected cubic graphs

2014 ◽  
Vol 06 (03) ◽  
pp. 1450032
Author(s):  
Shipeng Wang ◽  
Baoyindureng Wu ◽  
Xinhui An ◽  
Xiaoping Liu ◽  
Xingchao Deng
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Infinite Family ◽  
Connected Dominating Set ◽  
Minimum Size ◽  
Cubic Graphs ◽  
Longest Path ◽  
Connected Domination Number ◽  
Number Formula

A set S of vertices of a graph G is an outer-connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by V\S is connected. The outer-connected domination number [Formula: see text] is the minimum size of such a set. We present an infinite family of 2-connected cubic graphs, in which the number of vertices in a longest path are much less than the half of their orders. This disprove a recent conjecture posed by Akhbari, Hasni, Favaron, Karami, Sheikholeslami.

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