Efficient algorithms for tuple domination on co-biconvex graphs and web graphs

Abstract

A vertex in a graph dominates itself and each of its adjacent vertices. The $k$-tuple domination problem, for a fixed positive integer $k$, is to find a minimum sized vertex subset in a given graph such that every vertex is dominated by at least k vertices of this set. From the computational point of view, this problem is NP-hard. For a general circular-arc graph and $k=1$, efficient algorithms are known to solve it (Hsu et al., 1991 & Chang, 1998) but its complexity remains open for $k\geq 2$. A $0,1$-matrix has the consecutive 0's (circular 1's) property for columns if there is a permutation of its rows that places the 0's (1's) consecutively (circularly) in every column. Co-biconvex (concave-round) graphs are exactly those graphs whose augmented adjacency matrix has the consecutive 0's (circular 1's) property for columns. Due to A. Tucker (1971), concave-round graphs are circular-arc. In this work, we develop a study of the $k$-tuple domination problem on co-biconvex graphs and on web graphs which are not comparable and, in particular, all of them concave-round graphs. On the one side, we present an $O(n^2)$-time algorithm for solving it for each $2\leq k\leq |U|+3$, where $U$ is the set of universal vertices and $n$ the total number of vertices of the input co-biconvex graph. On the other side, the study of this problem on web graphs was already started by Argiroffo et al. (2010) and solved from a polyhedral point of view only for the cases $k=2$ and $k=d(G)$, where $d(G)$ equals the degree of each vertex of the input web graph $G$. We complete this study for web graphs from an algorithmic point of view, by designing a linear time algorithm based on the modular arithmetic for integer numbers. The algorithms presented in this work are independent but both exploit the circular properties of the augmented adjacency matrices of each studied graph class.Comment: 21 pages, 7 figures. Keywords: $k$-tuple dominating sets, augmented adjacency matrices, stable sets, modular arithmeti