Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

License no. 144 of season 1872/73 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 144 of season 1872/73 made out to S.D. Woodruff for 36 square miles in berth no. 192. This document is slightly torn and stained along the right hand side. This does not affect the text, April 7, 1873.

License no. 145 of season 1872/73 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 145 of season 1872/73 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, April 7, 1873.

License no. 67 of season 1873/74 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 67 of season 1873/74 made out to S.D. Woodruff for 36 square miles in berth no. 192, June 13, 1873

License no. 68 of season 1873/74 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 68 of season 1873/74 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, June 13, 1873.

License no. 11 of season 1874/75 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 11 of season 1874/75 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, May 20, 1874.

License no. 5 of season 1875/76 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 5 of season 1875/76 made out to S.D. Woodruff for 36 square miles in berth no. 192, June 1, 1875.

License no. 2 of season 1886/87 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 2 of season 1886/87 made out to S.D. Woodruff for 36 square miles in berth no. 192, May 15, 1876.

License no. 3 of season 1886/87 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 3 of season 1886/87 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, May 15, 1876

License no. 7 of season 1877/78 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198

License no. 7 of season 1877/78 made out to S.D. Woodruff for 35 ¾ square miles in berth no. 198, May 31, 1877.

License no. 6 of season 1877/78 made out to S.D. Woodruff for 36 square miles in berth no.192

License no. 6 of season 1877/78 made out to S.D. Woodruff for 36 square miles in berth no.192, May 31, 1877.

License no. 87 of season 1879/80 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 87 of season 1879/80 made out to S.D. Woodruff for 36 square miles in berth no. 192, June 3, 1879.

License no. 26 of season 1880/81 made out to S.D. Woodruff for 36 square miles in berth no. 192

License no. 26 of season 1880/81 made out to S.D. Woodruff for 36 square miles in berth no. 192, May 27, 1880.

Receipt from Morley’s Hotel, Trafalgar Square, London, England

Receipt from Morley’s Hotel, Trafalgar Square, London, England for meals, drinks and apartment, Aug. 13, 1847.