An Improved Upper Bound and Algorithm for Clique Covers

Indeterminate strings have received considerable attention in the recent past; see for example, the works of Helling et al and Christodoulakis et al. This attention is due to their applicability in bioinformatics, and to the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimum alphabet size of indeterminates representing any given undirected graph equals the size of the minimal clique cover of this graph. This paper first considers a related problem proposed by Helling el al: characterize the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph based on the vertex and edge counts. We provide improvements to the known upper bound, and a conjecture for the complete exact upper bound. Helling et al also present an algorithm which finds clique covers in polynomial time. We build on this result with a heuristic for vertex sorting which significantly improves their algorithm's results, particularly in dense graphs.