Total Graphs and Total Digraphs

Abstract

As investigation of the concept of a line graph has led to many interesting results, it is natural to expect that there has also been meaningful research into variations and generalizations of the subject. In this, the first of several chapters on such topics, instead of just the edges of a given graph becoming the vertices of a new graph, both the vertices and the edges of the original become vertices. The new graph is called the total graph of the original, and of course adjacency in the total graph has an appropriate definition, combining incidence and adjacency from the original graph. It so happens that total graphs have a nice alternative description in terms first of replacing the edges of a graph by paths of length 2 and then joining by an edge any vertices at distance 2. After some elementary properties of total graphs results on planarity, Eulerian and Hamiltonian total graphs are analyzed. The chapter concludes with results on total digraphs, which behave quite differently from total graphs.