A 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 \(\mathscr {O}(\log n)\) 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 \(\mathscr {O}(n^2d)\). This enables us to store the graph using \(\mathscr {O}(m\log n)\) bits of memory. For sparse graphs (graphs with \(\mathscr {O}(n)\) edges), this matches the trivial lower bound of \(\Omega (n\log n)\). As planar graphs and forests have constant degeneracy, our result implies an upper bound of \(\mathscr {O}(n^2)\) 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 \(\mathscr {O}(4^n)\), due to Kratochvíl, Miller & Nguyen (2001). Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time.

如果图 的顶点可以用不同的正整数来标记，并且当且仅当它们的标签之和是图的另一个顶点的标签时，两个顶点之间存在边，则该图被称为和图。大多数关于求和图的论文都考虑组合问题，例如需要添加到给定图以使其成为求和图的孤立顶点的最小数量。在本文中，我们从计算复杂度的角度开始对和图的研究。请注意，每个n顶点和图都可以由n 个正整数的排序列表表示，其中边查询可以在\(\mathscr {O}(\log n)\)中得到解答时间。因此，设置用​​作顶点标签的数字的上限也会设置数据库中存储图的空间复杂度的上限。我们证明，每个n顶点、m边、d简并图都可以通过添加至多m 个孤立顶点来构成和图，使得用作顶点标签的最大数字增长为\(\mathscr {O} (n^2d)\)。这使我们能够使用\(\mathscr {O}(m\log n)\)位内存来存储图形。对于稀疏图（具有\(\mathscr {O}(n)\)边的图），这与\(\Omega (n\log n)\)的平凡下界匹配。由于平面图和森林具有恒定的简并性，我们的结果意味着它们的标签数有\(\mathscr {O}(n^2)\)的上限。以前最著名的用最小数量的孤立顶点标记一般图所需的数字上限是\(\mathscr {O}(4^n)\)，由 Kratochvíl、Miller 和 Nguyen (2001) 提出。此外，他们的证明是存在性的，而我们的标签可以在多项式时间内构建。