Intro; Preface; Contents; 1 Introduction; 1.1 Overview; 1.2 Basic Graph Theoretical Definitions; 1.3 Specific Definitions and Some History; 1.4 Preliminary Results; 1.5 Applications; 2 General Results; 3 Connectivity Conditions; 4 Degree Conditions; 4.1 Minimum Degree; 4.2 Degree Sum; 4.3 Edge Density; 5 Domination Conditions; 6 Operations on Graphs; 6.1 Unary Operations; 6.1.1 Complement; 6.1.2 Line Graphs; 6.1.3 Permutation; 6.1.4 kth Power; 6.1.5 Iterated Corona; 6.2 Binary Operations; 6.2.1 Join; 6.2.2 Cartesian Product; 6.2.3 Direct Product; 7 Random Graphs

8 Proper k-Connection and Strong Proper Connection8.1 Proper k-Connection; 8.2 Strong Proper Connection; 8.2.1 General Results; 8.2.2 Operations on Graphs; 9 Proper Vertex Connection and Total Proper Connection; 9.1 Proper Vertex Connection; 9.2 Total Proper Connection; 10 Directed Graphs; 10.1 Proper Connection; 10.2 Proper Vertex Connection; 11 Other Generalizations; 11.1 (k, )-Rainbow Connectivity; 11.2 Proper Hamiltonian-Connection; 11.3 Proper-Walk Connection; 11.4 Proper Index; 11.5 Proper Distance; 11.6 Minimum Spanning Subgraphs; 12 Computational Complexity; References; Index