# Tuple domination on graphs with the consecutive-zeros property

## TL;DR

This paper investigates the computational complexity of the $k$-tuple domination problem on a special class of graphs characterized by the consecutive zeros property in their adjacency matrices, providing efficient algorithms for certain cases.

## Contribution

It introduces a polynomial-time algorithm for the $k$-tuple domination problem on graphs with the C0P property for columns, extending understanding of domination problems on circular-arc graphs.

## Key findings

- The $k$-tuple domination problem is solvable in linear time for graphs with the C0P property.
- The study extends the complexity analysis of domination problems to a new class of graphs.
- Efficient algorithms are provided for $2 \,\leq\, k \leq |U|+3$.

## 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 circular-arc graphs, its complexity remains open for $k\geq 2$. 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\leq k\leq |U|+3$, where $U$ is the set of universal vertices of $G$. From an algorithmic point of view, this takes linear time.