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 $O(\log n)$ time. Therefore, limiting the size of the 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 size of each vertex label is at most $O(n^2d)$. This enables us to store the graph using $O(m\log n)$ bits of memory. For sparse graphs (graphs with $O(n)$ edges), this matches the trivial lower bound of $\Omega(n\log n)$. Since planar graphs and forests have constant degeneracy, our result implies an upper bound of $O(n^2)$ on their label size. The previously best known upper bound on the label size of general graphs with the minimum number of isolated vertices was $O(4^n)$, due to Kratochv\'il, Miller & Nguyen. Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time.

翻译：如果它的顶点可以用明显的正数整数标注, 则图形就被称为一个总和图。 如果它的顶点总和是图中另一个顶点的标签标签, 并且只有两个顶点的总和是图中另一个顶点的标签。 因此, 对顶点标签的大小的限制也以存储数据库中的图形的空间复杂性为上限。 我们显示, 每一个$- verex, $- sedge, $- degenerate 图形都可以从计算复杂性的角度进行总和研究。 注意, 每一个 $ 的顶点总和图形都可以用一个美元正数的正数列表来表示。 注意, 每个美元 美元 的顶点总和数可以代表 $正数的正数整数, 边端查询以 $ (\ log n) 开始限制顶点标签的大小, 以美元 美元 。