Independent Dominating Set Research Articles

Independent Dominating Set on Chain of Fuzzy Graphs

In this paper, we applied some properties on chain fuzzy graphs, which comprise vertex identification. These properties are independent sets and independent dominant sets.

Domination and Independent Domination in Extended Supergrid Graphs

Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied the Hamiltonian-related properties of supergrid graphs and their subclasses of graphs. In this paper, we propose a generalized graph class for supergrid graphs called extended supergrid graphs. Extended supergrid graphs include grid graphs, supergrid graphs, diagonal supergrid graphs, and triangular supergrid graphs as subclasses of graphs. In this paper, we study the problems of domination and independent domination on extended supergrid graphs. A dominating set of a graph is the subset of vertices on it, such that every vertex of the graph is in this set or adjacent to at least a vertex of this set. If any two vertices in a dominating set are not adjacent, this is called an independent dominating set. Domination and independent domination problems find a dominating set and an independent dominating set with the least number of vertices on a graph, respectively. The domination and independent domination set problems on grid graphs are known to be NP-complete, meaning that these two problems on extended supergrid graphs are also NP-complete. However, the complexities of these two problems in other subclasses of graphs remain unknown. In this paper, we first prove that these two problems on diagonal supergrid graphs are NP-complete, then, by a simple extension, we prove that these two problems on supergrid graphs and triangular supergrid graphs are also NP-complete. In addition, these two problems on rectangular supergrid graphs are known to be linearly solvable; however, the complexities of these two problems on rectangular triangular-supergrid graphs remain unknown. This paper provides tight upper bounds on the sizes of the minimum dominating and independent dominating sets for rectangular triangular-supergrid graphs.

Survival backup strategy for controller placement problem in Software Defined Networking

The core concept of Software Defined Networking (SDN) is to abstract the control layer from the data layer. SDN architectures can provide programmatic interfaces in communication networks that significantly simplify network management and improve utilization efficiency. Distributed multiple controller SDN environments have become the preferred solution towards better scalability of SDNs. However, there are some inevitable problems for such networks, such as network failure, especially due to the failure of controllers themselves. To further improve network performance, reliability, and survivability, one solution is to deploy backup controllers in the network to satisfy quality of service requirements. In this paper, we aim to enhance the controller placement approach by designing a reliable and survivable controller placement framework. This framework consists of two levels. The first level is to determine the number and placement of the primary controllers using the node degree, which relies on selecting the nodes with the highest degree and then generating multiple domains exploiting the Independent Dominating Set (IDS) technique to ensure efficient distribution of controllers with lowest response times. The second level aims to provide some protection for the first level through the efficient deployment of backup controllers. The approach designed at this level is called the survivable backup controller placement approach. We evaluate this framework using real topologies; the obtained results show the significant impact of integrating both levels in improving the performance of SDNs without introducing unacceptable delay times.

Independent dominating set problem revisited

An independent dominating set of a graph G is a subset D of V such that every vertex not in D is adjacent to at least one vertex of D and no two vertices in D are adjacent. The independent dominating set (IDS) problem asks for an independent dominating set with minimum cardinality. First, we show that the independent dominating set problem and the dominating set problem on cubic bipartite graphs are both NP-complete. As an additional result, we give an alternative and more direct proof for the NP-completeness of both the independent dominating set problem and the dominating set problem on at-most-cubic grid graphs. Next, we show that there are fixed-parameter tractable algorithms for the independent dominating set problem and the dominating set problem on at-most-cubic graphs, which run in O(3.3028k+n) and O(4.2361k+n) time, respectively. Moreover, we consider the weighted independent dominating set problem on (k,ℓ)-graphs. We show that the problem on (2,1)-graphs is NP-complete. We also show that the problem can be solved in linear time for (1,1)-graphs and in polynomial time for (1,ℓ)-graphs for constant ℓ, respectively.

지배집합 알고리즘

This paper proposes a linear-time algorithm that has been designed to obtain an accurate solution for Dominating Set (DS) problem, which is known to be NP-complete due to the deficiency of polynomial-time algorithms that successfully derive an accurate solution to it. The proposed algorithm does so by repeatedly assigning vertex v with maximum degree among vertices adjacent to the vertex v with minimum degree to Minimum Independent DS (MIDS) as its element and removing all the incident edges until no edges remain in the graph. This algorithm finally transforms MIDS into Minimum DS (MDS) and again into Minimum Connected DS (MCDS) so as to obtain the accurate solution to all DS-related problems. When applied to ten different graphs, it has successfully obtained accurate solutions with linear time complexity O(n). It has therefore proven that Dominating Set problem is rather a P-problem.

Solving the Minimum Independent Domination Set Problem in Graphs by Exact Algorithm and Greedy Heuristic

In this paper we present a new approach to solve the Minimum Independent Dominating Set problem in general graphs which is one of the hardest optimization problem. We propose a method using a clique partition of the graph, partition that can be obtained greedily. We provide conditions under which our method has a better complexity than the complexity of the previously known algorithms. Based on our theoretical method, we design in the second part of this paper an efficient algorithm by including cuts in the search process. We then experiment it and show that it is able to solve almost all instances up to 50 vertices in reasonable time and some instances up to several hundreds of vertices. To go further and to treat larger graphs, we analyze a greedy heuristic. We show that it often gives good (sometimes optimal) results in large instances up to 60 000 vertices in less than 20 s. That sort of heuristic is a good approach to get an initial solution for our exact method. We also describe and analyze some of its worst cases.

Global vertex-edge domination chain and its characterization

In agraph G, a subset S of V is Global vertex-edge dominating set if S is vertex-edge dominating set in both G and . In this paper we have introduced new concepts such as Global vertex-edge dominating set, Global vertex-edge irredundant set, Global independent vertex-edge domination set. Also we have identified the Global vertex-edge domination chain, An attempt is made to characterize the Global vertex-edge domination number of graphs. We have identified the graphs whose Global vertex-edge domination number is 1, 2, 3 & k.

Clustering of wireless sensor and actor networks based on sensor distribution and connectivity

Wireless Sensor and Actor Networks (WSANs) employ significantly more capable actor nodes that can collect data from sensors and perform application specific actions. To take these actions collaboratively at any spot in the monitored regions, maximal actor coverage along with inter-actor connectivity is desirable. In this paper, we propose a distributed actor positioning and clustering algorithm which employs actors as cluster-heads and places them in such a way that the coverage of actors is maximized and the data gathering and acting times are minimized. Such placement of actors is done by determining the k-hop Independent Dominating Set (IDS) of the underlying sensor network. Basically, before the actors are placed, the sensors pick the cluster-heads based on IDS. The actors are then placed at the locations of such cluster-heads. We further derive conditions to guarantee inter-actor connectivity after the clustering is performed. If inter-connectivity does not exist, the actors coordinate through the underlying sensors in their clusters to adjust their locations so that connectivity can be established. The performances of the proposed approaches are validated through simulations.

Parameterized approximation of dominating set problems

A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k- Dominating Set, or (2) produces a dominating set of size at most g ( k ) , where g ( k ) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is no such FPT algorithm for g ( k ) of the form k + c (where c is a fixed constant, termed an additive FPT approximation), unless FPT = W [ 2 ] . We answer the analogous problem completely for the related Independent Dominating Set (IDS) problem, showing that IDS does not admit an FPT approximation algorithm, for any g ( k ) , unless FPT = W [ 2 ] .

Improved bottleneck domination algorithms

W.C.K. Yen introduced BOTTLENECK DOMINATION and BOTTLENECK INDEPENDENT DOMINATION. He presented an O ( n log n + m ) -time algorithm to compute a minimum bottleneck dominating set. He also obtained that the BOTTLENECK INDEPENDENT DOMINATING SET problem is NP-complete, even when restricted to planar graphs. We present simple linear time algorithms for the BOTTLENECK DOMINATING SET and the BOTTLENECK TOTAL DOMINATING SET problem. Furthermore, we give polynomial time algorithms (most of them with linear time-complexities) for the BOTTLENECK INDEPENDENT DOMINATING SET problem on the following graph classes: AT-free graphs, chordal graphs, split graphs, permutation graphs, graphs of bounded treewidth, and graphs of clique-width at most k with a given k-expression.

The weighted independent domination problem is NP-complete for chordal graphs

An independent dominating set of a graph G=( V, E) is a pairwise non-adjacent subset D of V such that every vertex not in D is adjacent to at least one vertex in D. Suppose each vertex in V is associated with a weight which is a real number. The weighted independent domination problem is to find an independent domination set of minimum total weights. This paper records an unpublished result of 20 years ago that the weighted independent domination problem is NP-complete for chordal graphs.

On linear and circular structure of (claw, net)-free graphs

We prove that every (claw, net)-free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present a linear time algorithm which, for a given (claw, net)-free graph, finds either a dominating pair or an induced doubly dominating cycle. We show also how one can use structural properties of (claw, net)-free graphs to solve efficiently the domination, independent domination, and independent set problems on these graphs.

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (G) n) time. The same technique can be used to show that the k -FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c 1 ^ \sqrt k n) time, where c 1 =3^ 36\sqrt 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k -DOMINATING SET, e.g., k -INDEPENDENT DOMINATING SET and k -WEIGHTED DOMINATING SET.

Independent Sets in Asteroidal Triple-Free Graphs

An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an O(n4) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain O(n4) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problems on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs.

Lower Ramsey numbers for graphs

Let μ( G) denote the smallest number of vertices in a maximal clique of the graph G, while i( G) (the independent domination number of G) denotes the smallest number of vertices in a maximal independent ( i.e. independent dominating) set of G. For given integers l and m, the lower Ramsey number s( l, m) originally defined in [4], is the largest integer p such that every graph G of order p has μ( G)⩽ l or i( G)⩽ m. We find an upper bound for s( l, m) which is better than the upper bound in [4] if l<⌊ m/2⌋. Combining this upper bound with a lower bound determined in [3], the numbers s(1, m) are determined exactly.

Domination in convex and chordal bipartite graphs

We state that various domination problems are polynomial time solvable in convex bipartite graphs, and we give the main ideas of the algorithms. TOTAL DOMINATING SET is polynomial even for chordal bipartite graphs. Further, it is shown by a reduction from 3SAT that INDEPENDENT DOMINATING SET remains NP-complete when restricted to chordal bipartite graphs.