Mixed covering arrays on graphs

Abstract

AbstractCovering arrays have applications in software, network and circuit testing. In this article, we consider a generalization of covering arrays that allows mixed alphabet sizes as well as a graph structure that specifies the pairwise interactions that need to be tested. Let k and n be positive integers, and let G be a graph with k vertices v1,v2,…, vk with respective vertex weights g1 ≤ g2 ≤ … ≤ gk. A mixed covering array on G, denoted by $CA( {n,G,\;\prod\nolimits_{i = 1}^k {g_i } } )$, is an n × k array such that column i corresponds to vi, cells in column i are filled with elements from ℤgi and every pair of columns i,j corresponding to an edge vi,vj in G has every possible pair from ℤgi × ℤgj appearing in some row. The number of rows in such array is called its size. Given a weighted graph G, a mixed covering array on G with minimum size is called optimal. In this article, we give upper and lower bounds on the size of mixed covering arrays on graphs based on graph homomorphisms. We provide constructions for covering arrays on graphs based on basic graph operations. In particular, we construct optimal mixed covering arrays on trees, cycles and bipartite graphs; the constructed optimal objects have the additional property of being nearly point balanced. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 393–404, 2007