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arXiv · 2510.02199

A polynomial algorithm to compute the boxicity and threshold dimension of complements of block graphs

Marco Caoduro·Will Evans·Tao Gaede
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Abstract

The boxicity of a graph $G$ is the minimum dimension $d$ that admits a representation of $G$ as the intersection graph of a family of axis-parallel boxes in $\mathbb{R}^d$. Computing boxicity is an NP-hard problem, and there are few known graph classes for which it can be computed in polynomial time. One such class is the class of block graphs. A block graph is a graph in which every maximal $2$-connected component is a clique. Since block graphs are known to have boxicity at most two, computing their boxicity amounts to the linear-time interval graph recognition problem. On the other hand, complements of block graphs have unbounded boxicity, yet we show that there is also a polynomial algorithm that computes the boxicity of complements of block graphs. An adaptation of our approach yields a polynomial algorithm for computing the threshold dimension of the complements of block graphs, which for general graphs is an NP-hard problem. Our method suggests a general technique that may show the tractability of similar problems on block-restricted graph classes.

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Marco Caoduro, Will Evans, Tao Gaede. 2025-10-02. A polynomial algorithm to compute the boxicity and threshold dimension of complements of block graphs. https://arxiv.org/abs/2510.02199

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