Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p ≥ 3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete. © 2008 Elsevier B.V. All rights reserved.

Some results on open-edge and open mobile guarding of polygons and triangulations

This paper focuses on a variation of the Art Gallery problem that considers open-edge guards and open mobile-guards. A mobile guard can be placed on edges and diagonals of a polygon, and the ‘open’ prefix means that the endpoints of such an edge or diagonal are not taken into account for visibility purposes. This paper studies the number of guards that are sufficient and sometimes necessary to guard some classes of simple polygons for both open-edge and open mobile-guards. A wide range of polygons is studied, which include orthogonal polygons with or without holes, spirals, orthogonal spirals and monotone polygons. Moreover, this problem is also considered for planar triangulation graphs using open-edge guards.