Strong Isometric Dimension

The strong isometric dimension of a finite connected graph is the least integer for which its graph distance can be realized by the infinity norm on . Thus there must be a map of its vertices into such that

The same definition applies to a tree whose edges have positive lengths, using the sum of lengths along the unique path between two vertices. It concerns the metric on the vertices and does not require a geometric drawing of the edges.

For a tree with leaves, the dimension is at least , where is the ceiling function and is the base-2 logarithm. Equality holds for every tree with at most 31 leaves. Chalmers (2026) constructed a 32-leaf tree requiring six coordinates, for every assignment of positive edge lengths, disproving the proposed equality in general. The work used AI for code and exposition, with the mathematical argument checked by the author. Independent external review had not been reported as of Sep. 7, 2026.