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

TL;DR

The paper studies computing the co-boxicity and threshold co-dimension for complements of block graphs. It develops a polynomial-time algorithm based on a co-interval cover framework that uses maximal co-interval subgraphs called big ants and the block-cut tree to assemble a minimum cover. It establishes that for block graphs both co-boxicity and threshold co-dimension are computable in polynomial time and proves the bounds $co-box(G) ≤ co-dim_{TH}(G) ≤ 2 · co-box(G)$. The approach suggests a general framework that could extend to other block-restricted classes such as cactus graphs.

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.