Super oriented cycles in permutations.

The degree of dissimilarity between genome sequences of homologous species is a measure of the evolutionary distance between them. It serves as a metric in the construction of phylogenetic trees, which depict the evolutionary relationships and common ancestry among different species. Given two genome sequences, evolutionary distance is determined by estimating the number of global mutations that transform one sequence to the other. The computation of the evolutionary distance is done by modelling a genome with the corresponding permutation. Global rearrangement operations such as transposition that model a particular genomic mutation are studied by employing a combinatorial structure known as a cycle graph of the corresponding permutation. A cycle in a cycle graph that has odd length is called an odd cycle. In the context of the problem of sorting by transpositions (SBT), a valid 2-move is a transposition that increases the number of odd cycles in the cycle graph by two. A super oriented cycle (SOC) is an odd cycle C where C and one of the resultant cycles admit valid 2-moves. The minimum number of mutations required to transform a species S into a related species T is the distance from S to T under that mutation. Christie opined that characterizing SOCs will improve the lower bound of the transposition distance. We characterize super oriented cycles. Equivalent transformations on permutations like reduction and (g,b)-split preserve the transposition distance of a given permutation and map SBT to the corresponding SBT on a transformed simpler permutation. We introduce merge, a novel equivalent transformation. These results have applications in computing transposition and other distances between related species.

Huge Recurrent Mucoepidermoid Carcinoma of Parotid - a Rare Presentation.

Mucoepidermoid carcinoma is the commonest primary malignant salivary gland tumor in both children and adults and constitutes around 2.8-16% of all salivary gland tumors. There is a wide age range of 3 to 95 years with a mean age of 47 years. We reported a case of 85 years old female patient, who presented to the department of Surgical Oncology, Srinivasan Medical College and Hospital, Trichy, with history of right parotid swelling for past 25 years, who had undergone Superficial parotidectomy 10 years back, followed by recurrence of tumor with huge increase in size, around 20 cm in greatest dimension. Right Completion Radical Parotidectomy with Modified Radical neck dissection type 1 with Cervical Rotation Flap cover was done and histopathological examination of the surgical specimen confirmed as high grade Mucoepidermoid carcinoma. Following surgery, the patient's outcome is better, uneventful with periodic regular follow-up.