This book constitutes the thoroughly refereed proceedings of the 39th International Workshop on Graph Theoretic Concepts in Computer Science, WG 2013, held in Lubeck, Germany, in June 2013. The 34 revised full papers presented were carefully reviewed and selected from 61 submissions. The book also includes two abstracts. The papers cover a wide range of topics in graph theory related to computer science, such as structural graph theory with algorithmic or complexity applications; design and analysis of sequential, parallel, randomized, parameterized and distributed graph and network algorithms; computational complexity of graph and network problems; computational geometry; graph grammars, graph rewriting systems and graph modeling; graph drawing and layouts; random graphs and models of the web and scale-free networks; and support of these concepts by suitable implementations and applications.