3. Perfect Graphs and Comparability Graphs

The first use of graph theory is dated to 1735. Swiss mathematician and physicist, Leonhard Euler, formulated theorems and definitions related to graphs. Euler was inspired by a real problem regarding crossing bridges in Königsberg (east Prussia, also known as Królewiec, Kaliningrad). He tried to take a walk around the city in such a way that each of the seven bridges was crossed only once. Finally, Euler proved that it was impossible, simultaneously solving the puzzles with the application of graph theory. Nowadays, such a theory is used in numerous fields of science and practical approaches. This chapter presents notations and definitions related to the graph theory. Furthermore, perfect graphs and comparability graphs are introduced. New theorems, lemmas, and algorithms regarding recognition and coloring of comparability graphs are proposed.