Showing papers on "Dominating set published in 1982"

Dominating Sets in Chordal Graphs

Abstract: A set of vertices D is a dominating set for a graph if every vertex is either in D or adjacent to a vertex which is in D. We show that the problem of finding a minimum dominating set in a chordal graph is NP-complete, even when restricted to undirected path graphs, but exhibit a linear time greedy algorithm for the problem further restricted to directed path graphs. Streamlined to handle only trees, our algorithm becomes the algorithm of Cockayne, Goodman and Hedetniemi. An interesting parallel with graph isomorphism is pointed out.

Domination‐balanced graphs

Abstract: A set D of vertices in a graph is said to be a dominating set if every vertex not in D is adjacent to some vertex in D. The domination number β(G) of a graph G is the size of a smallest dominating set. G is called domination balanced if its vertex set can be partitioned into β(G) subsets so that each subset is a smallest dominating set of the complement G of G. The purpose of this paper is to characterize these graphs.

An Approximation Algorithm for the Maximum Independent Set Problem on Planar Graphs

Abstract: In this paper we consider the maximum independent set problem in which one would like to find a maximum set of independent (i.e., pairwise nonadjacent) vertices in a given graph. The problem is NP-complete, and still remains so even if we restrict ourselves to the class of planar graphs. It has been conjectured that there exist no polynomial-time exact algorithms for any NP-complete problems. We present a polynomial-time approximation algorithm for the maximum independent set problem on planar graphs. For a given planar graph having any number n of vertices, our algorithm finds, in $O(n\log n)$ time, an independent set that is necessarily larger in size than half a maximum independent set. Thus the absolute worst case ratio of our algorithm is greater than $\tfrac{1}{2}$.