Perfect edge domination and efficient edge domination in graphs

  • Authors:

  • Chin Lung Lu;Ming-Tat Ko;Chuan Yi Tang

  • Affiliations:

  • Institute of Information Science, Academia Sinica, Nankang, Taipei 115, Taiwan, ROC;Institute of Information Science, Academia Sinica, Nankang, Taipei 115, Taiwan, ROC;Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan 30043, ROC

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2002

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Abstract

Let G = (V,E) be a finite and undirected graph without loops and multiple edges. An edge is said to dominate itself and any edge adjacent to it. A subset D of E is called a perfect edge dominating set if every edge of E \ D is dominated by exactly one edge in D and an efficient edge dominating set if every edge of E is dominated by exactly one edge in D. The perfect (efficient) edge domination problem is to find a perfect (efficient) edge dominating set of minimum size in G. Suppose that each edge e is associated with a real number w(e) as its weight. Then, the weighted perfect (efficient) edge domination problem is to calculate a perfect (efficient) edge dominating set D such that the weight w(D) of D is minimum, where w(D)=Σe∈D w(e). In this paper, we show that the perfect (efficient) edge domination problem is NP-complete on bipartite (planar bipartite) graphs. Moreover, we present linear-time algorithms to solve the weighted perfect (efficient) edge domination problem on generalized series-parallel graphs and chordal graphs.