A Graph Polynomial Approach to Primitivity? F. Blanchet-Sadri1, Michelle Bodnar2, Nathan Fox3, and Joe Hidakatsu2 1 Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402{6170, USA 2 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109{1043, USA 3 Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854{8019, USA Abstract. Recently, Tittmann et al. introduced the subgraph component polynomial and showed that its power for distinguishing graphs is quite different from the power of other graph polynomials that appear in the literature such as the matching polynomial, the Tutte polynomial, the characteristic polynomial, the chromatic polynomial, etc. The subgraph component polynomial enumerates vertex induced subgraphs in a given undirected graph with respect to the number of components. We show the use of the subgraph component polynomial to count the number of primitive partial words of a given length over an alphabet of a fixed size, which leads to a method for enumerating such partial words. 1 Introduction Motivated by social and biological networks, Tittmann et al. [10] introduced the subgraph component polynomial Q(G; x; y) of an undirected graph G with n vertices as the bivariate generating function which counts the number of con- nected components in vertex induced subgraphs. More precisely, Q(G; x; y) = n n i j Σi=0Σj=0qij(G)x y , where qij(G) is the number of vertex induced subgraphs of G with exactly i vertices and j connected components. They related the subgraph component polynomial to other graph polynomials that appear in the literature such as the Tutte polynomial, the universal edge elimination polynomial, etc. (see, for instance, [8] for more information on graph polynomials). They showed several remarkable properties of the subgraph component polynomial, among them is their use to compute the so-called \residual connectedness reliability". They also showed that the problem of computing Q(G; x; y) is ]P -hard, but that it is fixed parameter tractable when restricting to graph classes that have bounded tree-width and to classes of bounded clique-width. Primitivity is a well-studied topic in combinatorics on words (see, for in- stance, [4]). It is well-known that the number of primitive words of a given length over an alphabet of a fixed size can be calculated using the M¨obiusfunc- tion [7, 9]. In this paper, we discuss the use of the above subgraph component ? This material is based upon work supported by the National Science Foundation under Grant No. DMS{1060775. 2 F. Blanchet-Sadri, M. Bodnar, N. Fox, and J. Hidakatsu polynomial to count the number Ph;k(n) of primitive partial words with h holes of length n over a k-letter alphabet. Research on primitive partial words was initiated by the first author in [1]. Partial words, also referred to as strings with don't-cares, may have some undefined positions or holes. In [2], formulas for h = 1 and h = 2 are given in terms of the formula for h = 0 and some bounds are provided for h > 2, but no exact formulas are given for h > 2. Here, we associate a graph Gn;P with any partial word of length n with period set P as follows: the vertices represent the positions 0; : : : ; n − 1 and the edges are the pairs fi; i + mpg, where 0 ≤ i < i + mp ≤ n − 1, m 2 Z, and p 2 P. It turns out that Ph;k(n) can be expressed in terms of the Q(Gn;P ; x; y)'s. The contents of our paper are as follows: In Section 2, we answer the ques- tion \How many holes can a primitive partial word of length n over a k-letter alphabet contain?". We show that this number can be expressed in terms of the large factors of n. (Note that this count has recently led to efficient algo- rithms for computing all primitively-rooted squares and runs in partial words [3].) In Section 3, we describe a general method for counting primitive partial words with the subgraph component polynomial, which leads to a method for the enumeration of primitive partial words. In Section 4, we discuss in particular non-primitive partial words of length pq, where p and q are distinct primes. In doing so, we give a framework for understanding partial words that are exactly p- periodic and exactly q-periodic without being 1-periodic (this relates to a variant of Fine and Wilf's periodicity theorem [5]). In Section 5, we further discuss the n n−h computation of Nh;k(n) = h k − Ph;k(n), the number of non-primitive par- tial words with h holes of length n over a k-letter alphabet, using the subgraph component polynomial. Finally in Section 6, we conclude with some remarks. We end this section by reviewing a few basic concepts on partial words. Let A be a non-empty finite set, or an alphabet. We consider a partial word w over A as a word over the enlarged alphabet A = A [ {}, where the additional character plays the role of an undefined position or a hole. For 0 ≤ i < n, the character at position i of w is denoted by w(i). If w(i) 2 A, then i is defined, otherwise i is a hole. A full word is a partial word with an empty set of holes. We denote by w[i::j) the factor of w that starts at position i and ends at position j − 1, and by jwj the length of w or the number of characters in w. If w1 and w2 are two partial words of equal length, then w1 is contained in w2, denoted by w1 ⊂ w2, if w1(i) = w2(i) for all defined positions i in w1. The greatest lower bound of w1 and w2, denoted by w1 ^ w2, is the maximal partial word contained in both w1 and w2, i.e, (w1 ^w2) ⊂ w1 and (w1 ^w2) ⊂ w2, and if w ⊂ w1 and w ⊂ w2, then w ⊂ (w1^w2). For example, abba^aaa = a. For a positive integer p, a partial word w has a period of p or w is p-periodic if for all positions i; j defined in w such that i ≡ j mod p, we have w(i) = w(j). A partial word has an exact period of p or is exactly p-periodic if it is p-periodic and p divides its length. A partial word w is primitive if there exists no full word v such that w ⊂ vi with i ≥ 2, equivalently, if there is no proper factor p of jwj such that w is p-periodic. Clearly, if w is primitive and w ⊂ w0, then w0 is primitive. A Graph Polynomial Approach to Primitivity 3 2 Maximizing Number of Holes in Primitive Words We define the set of large factors LF (n) of an integer n as the set of integers m such that m < n, m j n, and for t 6= m; t j n, we have m - t. For example, when n = 30 we have LF (30) = f6; 10; 15g. Clearly, the large factors of n come from dividing n by its prime factors. Proposition 1. Given a primitive partial word of length n which contains the maximum number of holes, set LF (n) = ff1; : : : ; fmg. For any non-hole posi- tions i and j, we have i − j = c1f1 + ··· + cmfm for some ci 2 Z. Moreover, the fewest number of non-holes which rule out all of the large factors of n as periods is jLF (n)j + 1. Proof. To rule out the large factor f1, we must use two non-holes i1; i2 and they must differ by a multiple of f1. Note that for each i and j, we have lcm(fi; fj) = n, so i1 and i2 rule out at most one large factor, i.e., f1. To rule out the next large factor f2, there must exist a non-hole position i3 that differs from i1 or i2, say i1, by some multiple of f2. Since i1 − i2 = c1f1 and i1 − i3 = c2f2 for some c1; c2 2 Z, we get i2 − i3 = i2 − i1 + i1 − i3 = −c1f1 + c2f2. As noted earlier, the addition of i3 cannot possibly rule out any large factor other than f2. Continue in this way until all large factors have been ruled out as periods. After ruling out the first large factor with two non-holes, jLF (n)j − 1 non-holes are required to rule out the remaining jLF (n)j − 1 large factors. This necessitates a total of jLF (n)j + 1 non-hole positions to rule out all large factors of n. ut Theorem 1. The maximum number of holes that a primitive partial word of length n over an arbitrary alphabet of at least two letters can contain, denoted τ(n), is τ(n) = n − jLF (n)j − 1. Moreover, the maximum number of holes a primitive word can contain can be achieved using a binary alphabet. Proof. We begin with a construction over the binary alphabet fa; bg which shows that this number of holes can always be achieved. Let the word w be defined as w(i) = a if i + 1 2 LF (n), w(i) = b if i = n − 1, and w(i) = otherwise. There are jLF (n)j a's and one b, leaving room for exactly n−(jLF (n)j+1) holes as desired. We observe that it is unnecessary to check for incompatibilities in smaller periods because for any factor q which divides an element p of LF (n), an incompatibility in a period of length p implies an incompatibility in a period of length q, so we need only check that all periods given by our large factor set do not occur in w.