Theta Palindromes in Theta Conjugates

A DNA string is a Watson-Crick (WK) palindrome when the complement of its reverse is equal to itself. The Watson-Crick mapping (Formula Presented) is an involution that is also an antimorphism. (Formula Presented)-conjugates of a word is a generalization of conjugates of a word that incorporates the notion of WK-involution (Formula Presented). In this paper, we study the distribution of palindromes and Watson-Crick palindromes, also known as (Formula Presented)-palindromes among both the set of conjugates and (Formula Presented)-conjugates of a word w. We also consider some general properties of the set (Formula Presented), i.e., the set of (Formula Presented)-conjugates of a word w, and characterize words w such that (Formula Presented), i.e., with the maximum number of elements in (Formula Presented). We also find the structure of words that have at least one (WK)-palindrome in (Formula Presented).