The Space Complexity of Sum Labelling

Abstract

AbstractA graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every n-vertex sum graph can be represented by a sorted list of n positive integers where edge queries can be answered in $$\mathscr {O}(\log n)$$ O ( log n ) time. Therefore, upper-bounding the numbers used as vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every n-vertex, m-edge, d-degenerate graph can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as $$\mathscr {O}(n^2d)$$ O ( n 2 d ) . This enables us to store the graph using $$\mathscr {O}(m\log n)$$ O ( m log n ) bits of memory. For sparse graphs (graphs with $$\mathscr {O}(n)$$ O ( n ) edges), this matches the trivial lower bound of $$\Omega (n\log n)$$ Ω ( n log n ) . As planar graphs and forests have constant degeneracy, our result implies an upper bound of $$\mathscr {O}(n^2)$$ O ( n 2 ) on their label numbers. The previously best known upper bound on the numbers needed for labelling general graphs with the minimum number of isolated vertices was $$\mathscr {O}(4^n)$$ O ( 4 n ) , due to Kratochvíl, Miller & Nguyen (2001). Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time.