New Self-Stabilizing Algorithms for Minimal Weakly Connected Dominating Sets

2015 ◽  
Vol 26 (02) ◽  
pp. 229-240
Author(s):  
Yihua Ding ◽  
James Z. Wang ◽  
Pradip K. Srimani
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Graph Algorithm ◽  
Connected Dominating Set ◽  
Dominating Sets ◽  
Time And Space ◽  
Stabilization Time ◽  
Connected Dominating Sets ◽  
Space Requirement ◽  
Weakly Connected Dominating Set

In this paper, we propose two new self-stabilizing algorithms, MWCDS-C and MWCDS-D, for minimal weakly connected dominating sets in an arbitrary connected graph. Algorithm MWCDS-C stabilizes in O(n4) steps using an unfair central daemon and space requirement at each node is O(log n) bits at each node for an arbitrary connected graph with n nodes; it uses a designated node while other nodes are identical and anonymous. Algorithm MWCDS-D stabilizes using an unfair distributed daemon with identical time and space complexities, but it assumes unique node IDs. In the literature, the best reported stabilization time for a minimal weakly connected dominating set algorithm is O(nmA) under a distributed daemon [1], where m is the number of edges and A is the number of moves to construct a breadth-first tree.

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.

Doubly connected domination in the join and Cartesian product of some graphs

2014 ◽  
Vol 07 (04) ◽  
pp. 1450054
Author(s):  
Benjier H. Arriola ◽  
Sergio R. Canoy
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Cartesian Product ◽  
Connected Dominating Set ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Restrained Domination ◽  
Total Restrained Domination ◽  
Positive Integers ◽  
Simple Connected Graph

Let G be a simple connected graph. A connected dominating set S ⊂ V(G) is called a doubly connected dominating set of G if the subgraph 〈V(G)\S〉 induced by V(G)\S is connected. We show that given any three positive integers a, b, and c with 4 ≤ a ≤ b ≤ c, where b ≤ 2a, there exists a connected graph G such that a = γr(G), b = γtr(G), and c = γcc(G), where γr, γtr, and γcc are, respectively, the restrained domination, total restrained domination, and doubly connected domination parameters. Also, we characterize the doubly connected dominating sets in the join of any graphs and Cartesian product of some graphs. The corresponding doubly connected domination numbers of these graphs are also determined.

Some results for the two disjoint connected dominating sets problem

2019 ◽  
Vol 11 (06) ◽  
pp. 1950065
Author(s):  
Xianliang Liu ◽  
Zishen Yang ◽  
Wei Wang
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Sufficient Condition ◽  
Dominating Sets ◽  
Cubic Graph ◽  
Connected Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Dominating Set Problem

As a variant of minimum connected dominating set problem, two disjoint connected dominating sets (DCDS) problem is to ask whether there are two DCDS [Formula: see text] in a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], and if not, how to add an edge subset with minimum cardinality such that the new graph has a pair of DCDS. The two DCDS problem is so hard that it is NP-hard on trees. In this paper, if the vertex set [Formula: see text] of a connected graph [Formula: see text] can be partitioned into two DCDS of [Formula: see text], then it is called a DCDS graph. First, a necessary but not sufficient condition is proposed for cubic (3-regular) graph to be a DCDS graph. To be exact, if a cubic graph is a DCDS graph, there are at most four disjoint triangles in it. Next, if a connected graph [Formula: see text] is a DCDS graph, a simple but nontrivial upper bound [Formula: see text] of the girth [Formula: see text] is presented.

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 Minimum Connected Dominating Sets of Random Cubic Graphs

10.37236/1624 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
W. Duckworth
Keyword(s):  
Differential Equations ◽  
Greedy Algorithm ◽  
Dominating Set ◽  
Connected Dominating Set ◽  
Cubic Graphs ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Average Case ◽  
Simple Heuristic

We present a simple heuristic for finding a small connected dominating set of cubic graphs. The average-case performance of this heuristic, which is a randomised greedy algorithm, is analysed on random $n$-vertex cubic graphs using differential equations. In this way, we prove that the expected size of the connected dominating set returned by the algorithm is asymptotically almost surely less than $0.5854n$.

scholarly journals New Computation of Resolving Connected Dominating Sets in Weighted Networks

Entropy ◽  
2019 ◽  
Vol 21 (12) ◽  
pp. 1174
Author(s):  
Adriana Dapena ◽  
Daniel Iglesia ◽  
Francisco J. Vazquez-Araujo ◽  
Paula M. Castro
Keyword(s):  
Complex Networks ◽  
Dominating Set ◽  
Connected Dominating Set ◽  
Virtual Backbone ◽  
Dominating Sets ◽  
Connected Subset ◽  
Resolving Set ◽  
Connected Dominating Sets ◽  
Minimum Cardinality ◽  
Information Interchange

In this paper we focus on the issue related to finding the resolving connected dominating sets (RCDSs) of a graph, denoted by G. The connected dominating set (CDS) is a connected subset of vertices of G selected to guarantee that all vertices in the graph are connected to vertices in the CDS. The connected dominating set with minimum cardinality, or minimum CDS (MCDS), is an adequate virtual backbone for information interchange in a network. When distinct vertices of G have also distinct representations with respect to a subset of vertices in the MCDS, it is said that the MCDS includes a resolving set (RS) of G. With this work, we explore different strategies to find the RCDS with minimum cardinality in complex networks where the vertices have different importances.

A ZONAL ALGORITHM FOR CLUSTERING AN HOC NETWORKS

2003 ◽  
Vol 14 (02) ◽  
pp. 305-322 ◽  
Author(s):  
YUANZHU PETER CHEN ◽  
ARTHUR L. LIESTMAN
Keyword(s):  
Ad Hoc ◽  
Mobile Ad Hoc Network ◽  
Dominating Set ◽  
Connected Dominating Set ◽  
Dominating Sets ◽  
New Approach ◽  
Graph Domination ◽  
Np Complete ◽  
Hoc Networks ◽  
Weakly Connected Dominating Set

A Mobile Ad Hoc Network (MANET) is an infrastructureless wireless network that can support highly dynamic mobile units. The multi-hop feature of a MANET suggests the use of clustering to simplify routing. Graph domination can be used in defining clusters in MANETs. A variant of dominating set which is more suitable for clustering MANETs is the weakly-connected dominating set. A cluster is defined to be the set of vertices dominated by a particular vertex in the dominating set. As it is NP-complete to determine whether a given graph has a weakly-connected dominating set of a particular size, we present a zonal distributed algorithm for finding small weakly-connected dominating sets. In this new approach, we divide the graph into regions, construct a weakly-connected dominating set for each region, and make adjustments along the borders of the regions to produce a weakly-connected dominating set of the entire graph. We present experimental evidence that this zonal algorithm has similar performance to and provides better cluster connectivity than previous algorithms.

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 $k=1$.

Double hop dominating sets in graphs

2020 ◽  
pp. 2150057
Author(s):  
Reynaldo V. Mollejon ◽  
Sergio R. Canoy
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Binary Operations

Let [Formula: see text] be a connected graph of order [Formula: see text]. A subset [Formula: see text] is a double hop dominating set (or a double [Formula: see text]-step dominating set) if [Formula: see text], where [Formula: see text], for each [Formula: see text]. The smallest cardinality of a double hop dominating set of [Formula: see text], denoted by [Formula: see text], is the double hop domination number of [Formula: see text]. In this paper, we investigate the concept of double hop dominating sets and study it for graphs resulting from some binary operations.

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