Efficient Parallel Algorithms on Distance-Hereditary Graphs (Extended Abstract) z Sun-yuan Hsieh zyxwvu * Chin-Wen Ho Dept. of Computer Science & Info. Eng. National Taiwan University, Taiwan e-mail: d3506013 @csie.ntu.edu.tw e-mail: hocw@csie.ncu.edu.tw Tsan-sheng Hsu t Ming-Tat KO Dept. of Computer Science & Info. Eng. National Central University, Taiwan Institute of Information Science Institute of Information Science Academia Sinica, Taiwan Academia Sinica, Taiwan e-mail: tshsu@iis.sinica.edu.tw e-mail: mtko@iis.sinica.edu.tw Gen-Huey Chen Dept. of Computer Science & Info. Eng. National Taiwan University, Taiwan e-mail: ghchen@csie.ntu.edu.tw Abstract zyxwvutsr In this papel; we present eficient parallel algorithms forjinding a minimum weighted connected dominating set, a minimum weighted Steiner tree for a distance-hereditary graph which take O(1og n ) time using O(n+m) processors on a CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively. We also find a maximum weighted clique zyxwvutsr of a distance-hereditary - - graph in O(log2 n) time using O(n + m) processors on a CREW PRAM. 1 Introduction A graph is distance-hereditary if every two vertices have the same distance in every connected induced subgraph containing both (where the distance between two vertices is the length of a shortest path connecting them). Properties and optimization problems in distance-hereditary graphs have been extensively studied during the past two decades [2,7, 8,9, 111. A dominating set D of a graph G = (V, zyxwvu E) is defined as D V such that every vertex in V \ D is either in D or is adjacent to some vertex in D. A dominating set D is connected if the subgraph induced by D is connected. For a given graph G and a set K zyxwvut C V (of terminal ver- tices), a Steiner tree is a tree which spans all vertices of I<-. The connected dominating set problem CD (respectively, Steiner tree problem zyxwvuts S7) asks for a minimum cardinality connected dominating set (respectively, Steiner tree). In this paper, we consider the following weighted version of the CD and ST on distance-hereditary graphs. Let w(w) be a non-negative weight associated with each vertex w in 'Supported in part by Institute of Information Science, Pcademia Sinica, Teipei, Taiwan. Supported in part by NSC Grant zyxwvutsrqpon 86-22 13-E-001-012, G. We want to find a connected dominating set D (respec- tively, Steiner tree T) such that xwED w(w) (respectively, CwET w(w)) equals the smallest possible value. We show that the above problems can be solved in O(1ogn) time using O(n + m) processors on a CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively. A graph C is a clique if there is an edge between every pair of vertices. We say that C is a clique of G if C is an in- duced clique of G. If each vertex is assigned anon-negative weight, a maximum weighted clique is a clique of G with the maximum total weight. In this paper, we also compute a maximum weighted clique of a distance-hereditary graph in O(log2 n) time using O(n + m) processors on a CREW PRAM. 2 Preliminaries This paper considers finite, simple loopless, undirected and connected graphs G = (V, E), where V and E are the vertex and edge set of G, respectively. Let n = IVI and m = (El. The distance &(z, y) or d(z, y) between two vertices z and y in G is the length of a shortest z-y path in G. Let v be a vertex of G. We denote the neighborhood of w, consisting of all vertices adjacent to w, by N(w), and the closed neighborhood of w, the set N(w) U {w}, by N[w]. Let S be a subset of V. We denote N(S) the open neighborhood of S, that is the set of vertices in G, exclusive of S, which are adjacent to any vertex in S. We also denote N[S] = N(S) U S. The subgraph induced by S, denoted by (S), consists of the vertices of S and edges (2, y) with 2, y E S and (z, y) E E. The hanging of a connected graph G = (Vi E) at a vertex U E V, denoted as h,, is the collection of sets &(U), Ll(u) ,..., Lt(u)(orsimplyLo,L1 ,..., Lt whennoambiguity 20 0190-3918/97 $10.00 0 1997 IEEE