An improved upper bound and algorithm for clique covers

Indeterminate strings have received considerable attention in the recent past; see for example [1] and [3]. 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 in [3]: characterize Θn(m), which is 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 on n vertices and m edges. We provide improvements to the known upper bound for Θn(m) in section 3.3. [3] also presents 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.