Tuple Domination on Graphs with the Consecutive-zeros Property

Abstract

The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The k-tuple domination is NP-hard even for chordal graphs. For the class of circulararc graphs, its complexity remains open for k ≥ 2. A 0, 1-matrix has the consecutive 0's property (C0P) for columns if there is a permutation of its rows that places the 0's consecutively in every column. Due to A. Tucker, graphs whose augmented adjancency matrix has the C0P for columns are circular-arc. In this work we study the k-tuple domination problem on graphs G whose augmented adjacency matrix has the C0P for columns, for 2 ≤ k ≤ |U | + 3, where U is the set of universal vertices of G. From an algorithmic point of view, this takes linear time.