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Let G = (V, E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhood N [v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v ∈ V , there is exactly one d ∈ D dominating v. An edge set M ⊆ E is an efficient edge dominating (e.e.d.) set for G if it is an(More)
Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted(More)
The class L k of k-leaf powers consists of graphs G = (V, E) that have a k-leaf root, that is, a tree T with leaf set V , where xy ∈ E, if and only if the T-distance between x and y is at most k. Structure and linear time recognition algorithms have been found for 2-, 3-, 4-, and, to some extent, 5-leaf powers, and it is known that the union of all k-leaf(More)
Polar and monopolar graphs are natural generalizations of bipartite or split graphs. A graph G = (V, E) is polar if its vertex set admits a partition V = A∪B such that A induces a complete multipartite and B the complement of a complete multipartite graph. If A is even a stable set then G is called monopolar. Recognizing general polar or monopolar graphs is(More)
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