Mixed covering arrays on graphs of small treewidth

Abstract

Covering arrays are combinatorial objects that have been successfully applied in design of test suites for testing systems such as software, hardware, and networks where failures can be caused by the interaction between their parameters. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text]. Two vectors [Formula: see text] and [Formula: see text] are qualitatively independent if for any ordered pair [Formula: see text], there exists an index [Formula: see text] such that [Formula: see text]. Let [Formula: see text] be a graph with [Formula: see text] vertices [Formula: see text] with respective vertex weights [Formula: see text]. A mixed covering array on[Formula: see text] , denoted by [Formula: see text], is a [Formula: see text] array such that row [Formula: see text] corresponds to vertex [Formula: see text], entries in row [Formula: see text] are from [Formula: see text]; and if [Formula: see text] is an edge in [Formula: see text], then the rows [Formula: see text] are qualitatively independent. The parameter [Formula: see text] is the size of the array. Given a weighted graph [Formula: see text], a mixed covering array on [Formula: see text] with minimum size is optimal. In this paper, we introduce some basic graph operations to provide constructions for optimal mixed covering arrays on the family of graphs with treewidth at most three.