The kdomination number of a graph G, y k ( G ) , is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k. thenis the smallest cardinality of a k-dominating set of G. The theory of k-domination in graphs has been extensively studied by Fink and Jacobson [l]. Our contribution is to prove the following result. Proof. Suppose that the result is false and that G is a counterexample having the least number of edges. If r = p -yk(G) then, by assumption, and we deduce x ( G ) > kr. Notice that for any independent set U of vertices of G, V -U is k-dominating.By minhality of G, the set S = {u E VI d(u) > k} is independent (or empty).Let T be a maximal independent set containing S. Then V -T is k-dominating and
Abstract. In this paper we investigate the complexity of finding maximum right angle free subsets of a given set of poi1.1-ts in the plane. For a set of rational points P in the plane. the right angle number p(P) (respectively rectilmear right angle number P.R(P)) of P is the cardinality of a maximum subset of P, no three members of which form a right angle triangle (respectively a right angle triangle with its side or base parallel to the x-axis~ It is shown that both parameters are .¥9'-hard to compute. The latter problem is also shown to be equivalent to finding a minimum dominating set in a bipartite graph. This is used to show that there is a polynomial algorithm for computing PB.(P) when P is a horizontally-convex subset of the lattice Z x Z (P is horizontally-convex if for any pair of points in P which lie on a horizontal line, every lattice point between them is also in P). We then show that this algorithm yields a !-approximate algorithm for the right angle number of a convex subregion of the lattice. t. IntroductionLet P be a set of points in the plane. A subset S of P is right angle free (nlf) if no three points of S form the vertices of a right angle triangle. The right angle number of P, denoted by p(P), is the cardinality of a maximum raf subset of P.
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