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Convex and isometric domination of (weak) dominating pair graphs

Research paper by Boštjan Brešar, Tanja Gologranc; Tim Kos

Indexed on: 03 Jun '18Published on: 28 May '18Published in: Theoretical Computer Science



Abstract

Publication date: 19 June 2018 Source:Theoretical Computer Science, Volume 730 Author(s): Boštjan Brešar, Tanja Gologranc, Tim Kos A set D of vertices in a graph G is a dominating set if every vertex of G, which is not in D, has a neighbor in D. A set of vertices D in G is convex (respectively, isometric), if all vertices in all shortest paths (respectively, all vertices in one of the shortest paths) between any two vertices in D lie in D. The problem of finding a minimum convex dominating (respectively, isometric dominating) set is considered in this paper from algorithmic point of view. For the class of weak dominating pair graphs (i.e., the graphs that contain a dominating pair, which is a pair of vertices x , y ∈ V ( G ) such that vertices of any path between x and y form a dominating set), we present an efficient algorithm that finds a minimum isometric dominating set of such a graph. On the other hand, we prove that even if one restricts to weak dominating pair graphs that are also chordal graphs, the problem of deciding whether there exists a convex dominating set bounded by a given arbitrary positive integer is NP-complete. By further restricting the class of graphs to chordal dominating pair graphs (i.e., the chordal graphs in which every connected induced subgraph has a dominating pair) we are able to find a polynomial time algorithm that determines the minimum size of a convex dominating set of such a graph.