International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 62-73

# Border Correlations, Lattices, and the Subgraph Component Polynomial

• Michelle Cordier
• Rachel Kirsch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

## Abstract

We consider the border sets of partial words and study the combinatorics of specific representations of them, called border correlations, which are binary vectors of same length indicating the borders. We characterize precisely which of these vectors are valid border correlations, and establish a one-to-one correspondence between the set of valid border correlations and the set of valid period correlations of a given length, the latter being ternary vectors representing the strong and strictly weak period sets. It turns out that the sets of all border correlations of a given length form distributive lattices under suitably defined partial orderings. We also investigate the population size, i.e., the number of partial words sharing a given border correlation, and obtain formulas to compute it. We do so using the subgraph component polynomial of an undirected graph, introduced recently by Tittmann et al. (European Journal of Combinatorics, 2011), which counts the number of connected components in vertex induced subgraphs.

## Keywords

Undirected Graph Partial Word Cycle Graph Binary Alphabet Graph Polynomial
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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