Series-Parallel Operations with Alpha-Graphs

Abstract

Among difference vertex labelings of graphs, α-labelings are the most restrictive one. A graph is an α-graph if it admits an α-labeling. In this work, we study a new alternative to construct α-graphs using, the well-known, series-parallel operations on smaller α-graphs. As an application of the series operation, we show that all members of a subfamily of all trees with maximum degree 4, obtained using vertex amalgamation of copies of the path P11, are α-graphs. We also show that the one-point union of up to four copies of Kn,n is an α-graph. In addition we prove that any α-graph of order m and size n is an induced subgraph of a graph of order m+ 2 and size m+ n. Furthermore, we prove that the Cartesian product of the bipartite graph K2,n and the path Pm is an α-graph.