International Journal of Computer and Information Sciences, VoI. 10, No. 2, 1981 Minimum Dominating Cycles in Outerplanar Graphs Andrzej Proskurowski 1 and Maciej M. Systo 2 Received August 1980; revised May 1981 A dominating cycle of a graph lies at a distance of at most one from all the vertices of the graph. The problem of finding the minimum size of such a cycle is proved to be difficult even when restricted to planar graphs. An efficient algorithm solving this problem is given for the class of two-c~nnected outer- planar graphs, in which all vertices lie on the exterior face in a plane embedding of the graph. KEY WORDS: Domination; NP-completeness; Outerplanar graphs. 1. INTRODUCTION A notion of domination in graphs is closely related to the problem of distributing resources in networks. A subset D of vertices of a graph G = (V, E), D ___ V, is said to dominate the set of vertices V if any vertex not in D is adjacent to a vertex in D. In this paper we consider a dominating cycle of a graph. A dominating cycle is a dominating set of vertices vo,v 1..... v k of the graph which form a cycle (vertices v i and Vi+l are adjacent, for 0 ~< i ~< k and addition modulo k + 1). Let us define the size of a graph as the number of its edges. We are interested in the problem of algorithmically determining a minimum dominating cycle of a given graph G, i.e., a cycle of minimum size among all dominating cycles of G. We will show that this problem is NP-complete for planar graphs, which calls for investigation of some more restricted classes of graphs. In Ref. 3, a solution has been proposed for a class of graphs including maximal outerplanar graphs, which are triangulations of a polygon. There the reader may also 1 Department of Computer and Information Science, University of Oregon, Eugene, Oregon. 2 Computer Science Department, Washington State University, Pullman, Washington (on leave from Institute of Computer Science, University of Wroclaw, Wroctaw, Poland). 127 0091-7036/81/0400-0127503.00/0 9 1981 Plenum Publishing Corporation