Mixed covering arrays on graphs of small treewidth

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 n and k be positive integers with k ≥ 3. Two vectors x ∈ Zng1 and y ∈ Zng2 are qualitatively independent if for any ordered pair (a,b) ∈ Zg1 × Zg2, there exists an index j ∈ {0, 1,.,n-1} such that (x(j),y(j)) = (a,b). Let G be a graph with k vertices v1,v2,.,vk with respective vertex weights g1,g2,.,gk. A mixed covering array onG, denoted by CA(n,G, Πki=1kg i), is a k × n array such that row i corresponds to vertex vi, entries in row i are from Zgi; and if {vx,vy} is an edge in G, then the rows x,y are qualitatively independent. The parameter n is the size of the array. Given a weighted graph G, a mixed covering array on G 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.