Testing whether a subgraph is convex or isometric

TL;DR

This paper investigates the computational complexity of determining whether a subgraph is convex or isometric within a larger graph, providing algorithms and lower bounds for various graph classes.

Contribution

It introduces new algorithms with improved running times for specific graph classes and establishes conditional lower bounds for sparse graphs.

Findings

Subquadratic algorithms for planar graphs.

Near-linear algorithms for graphs of bounded treewidth.

Conditional lower bounds for sparse graphs.

Abstract

We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved by computing the distances between all pairs of vertices. We provide a conditional lower bound showing that, for sparse graphs with $n$ vertices and $\Theta(n)$ edges, we cannot expect to solve the problem in $O(n^{2-\varepsilon})$ time for any constant $\varepsilon>0$. We also show that the problem can be solved in subquadratic time for planar graphs and in near-linear time for graphs of bounded treewidth. Finally, we provide a near-linear time algorithm for the setting where $G$ is a plane graph and $H$ is defined by a few cycles in $G$.