Isometric-Universal Graphs for Trees

TL;DR

This paper studies the minimal graphs that can embed two given trees isometrically, providing algorithms with polynomial time complexity and proving limitations for extending these results to more than two forests.

Contribution

The paper introduces efficient algorithms for finding smallest isometric-universal graphs for two trees and extends the analysis to forests, while establishing complexity and structural limitations.

Findings

Polynomial-time algorithms for two trees

Extension to forests with higher complexity

NP-completeness for three forests

Abstract

We consider the problem of finding the smallest graph that contains two input trees each with at most $n$ vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for two given trees. We prove that this problem can be solved in time $O(n^{5/2}\log{n})$. We extend this result to forests instead of trees, and propose an algorithm with running time $O(n^{7/2}\log{n})$. As a key ingredient, we show that a smallest isometric-universal graph of two trees essentially is a tree. Furthermore, we prove that these results cannot be extended. Firstly, we show that deciding whether there exists an isometric-universal graph with $t$ vertices for three forests is NP-complete. Secondly, we show that any smallest isometric-universal graph cannot be a tree for some families of three trees. This latter result has implications for greedy strategies solving the smallest isometric-universal graph problem.