A note on block-and-bridge preserving maximum common subgraph algorithms for outerplanar graphs

TL;DR

This paper critically examines a polynomial-time algorithm for block-and-bridge preserving maximum common subgraph in outerplanar graphs, revealing errors in its claimed efficiency and properties, and discusses potential improvements.

Contribution

It identifies inaccuracies in the original algorithm's complexity analysis and disproves the metric property of the derived dissimilarity measure, providing insights for future improvements.

Findings

The algorithm's actual complexity is at least O(n^4) for general cases.

The dissimilarity measure does not satisfy the triangle inequality.

Suggestions are provided for potential algorithm enhancements.

Abstract

Schietgat, Ramon and Bruynooghe proposed a polynomial-time algorithm for computing a maximum common subgraph under the block-and-bridge preserving subgraph isomorphism (BBP-MCS) for outerplanar graphs. We show that the article contains the following errors: (i) The running time of the presented approach is claimed to be $\mathcal{O}(n^{2.5})$ for two graphs of order $n$. We show that the algorithm of the authors allows no better bound than $\mathcal{O}(n^4)$ when using state-of-the-art general purpose methods to solve the matching instances arising as subproblems. This is even true for the special case, where both input graphs are trees. (ii) The article suggests that the dissimilarity measure derived from BBP-MCS is a metric. We show that the triangle inequality is not always satisfied and, hence, it is not a metric. Therefore, the dissimilarity measure should not be used in combination with techniques that rely on or exploit the triangle inequality in any way. Where possible, we give hints on techniques that are suitable to improve the algorithm.