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This dissertation would not be possible without the assistance of my committee members, my family, and my friends. Foremost, I am grateful to my advisor, Dr. Bogdan Oporowski. He always encouraged me to move forward. I also owe my deepest gratitude to Dr. James Oxley. He advised me to continue working with patience and sharpness for my research. I am also grateful to Dr. Guoli Ding. He introduced me Combinatorica, a package of Mathematica for my work. Addition-ally, I also would like to show my gratitude to Dr. Stan Dziobiak. He helped me how to operate the program. This work was motivated by Theorem 3.4 in [DOTV11] brought to my attention by the similarity between crossing number and planarity. I am indebted to my family members. They always support me and show an unconditional source of love throughout my life. Finally, I appreciate my friends cross the world. I am very fortunate to have had their friendship. I gratefully acknowledge the support from all people who supported me to com