Difference Labeling and Decomposition

A difference labeling of a graph G is an injective function f : V (G) → N ∪ {0} together with the weight function f∗ on E(G) given by f∗(uv) = |f(u) − f(v)| for every edge uv in G. The collection of subgraphs induced by the edges of the same weight is a decomposition of G and is called the common weight decomposition of G induced by f. Let ψf denote the collection of all the paths taken from each member of the common weight decomposition induced by f. A difference labeling f of G is said to be a graphoidal difference labeling if ψf is an acyclic graphoidal decomposition of G. This paper initiates a study on this concepts.