Showing papers on "Dominating set published in 1984"
TL;DR: It is shown that the problem of finding a minimum cardinality dominating set is NP-complete for split graphs and bipartite graphs.
178 citations
TL;DR: For undirected path graphs, the problem remains NP-complete as discussed by the authors, and it was shown that the problem is not solvable in linear time for any tree, even in the case of trees.
Abstract: A set of vertices D is a dominating set for a graph $G = (V,E)$ if every vertex not in D is adjacent to a vertex in D. A set of vertices is a total dominating set if every vertex in V is adjacent to a vertex in D. Cockayne, Goodman and Hedetniemi presented a linear time algorithm to determine minimum dominating sets for trees. Booth and Johnson established the NP-completeness of the problem for undirected path graphs. This paper presents a linear time algorithm to determine minimum total dominating sets of a tree and shows that for undirected path graphs the problem remains NP-complete.
106 citations
TL;DR: It is shown that questions concerning the complexity of approximation algorithms for the Maximum Independent Set and Minimum Vertex Cover Problems are at least as hard as a sample of other open questions concerning approximating NP-hard problems, including Graph Coloring, Set Covering and Dominating Set Problems.
43 citations
Journal Article•
10 citations
TL;DR: A backtracking algorithm and a dynamic programming algorithm to determine a minimum independent dominating set and it is shown that the algorithm is effective.
Abstract: The problem of determining a minimum independent dominating set is fundamental to both the theory and applications of graphs. Computationally it belongs to the class of hard combinatorial optimization problems known as NP-hard. In this paper, we develop a backtracking algorithm and a dynamic programming algorithm to determine a minimum independent dominating set. Computational experience with the backtracking algorithm on more than 1000 randomly generated graphs ranging from 100 to 200 vertices and from 10% to 60% densities has shown that the algorithm is effective.
2 citations