# Distance-based (and path-based) covering problems for graphs of given cyclomatic number

## TL;DR

This paper investigates graph covering problems related to distances in graphs with bounded cyclomatic number, providing new bounds, algorithms, and solutions for various problems including metric dimension, geodetic sets, and path covers.

## Contribution

It generalizes previous bounds, improves results, and introduces a linear-time algorithmic technique for solving these problems in graphs with bounded cyclomatic number.

## Key findings

- Optimal values are bounded by a linear function of cyclomatic number and degree-1 vertices.
- Provides near-optimal bounds and solves some open conjectures.
- Algorithms run in linear time for graphs with bounded cyclomatic number.

## Abstract

We study a large family of graph covering problems, whose definitions rely on distances, for graphs of bounded cyclomatic number (that is, the minimum number of edges that need to be removed from the graph to destroy all cycles). These problems include (but are not restricted to) three families of problems: (i) variants of metric dimension, where one wants to choose a small set $S$ of vertices of the graph such that every vertex is uniquely determined by its ordered vector of distances to the vertices of $S$; (ii) variants of geodetic sets, where one wants to select a small set $S$ of vertices such that any vertex lies on some shortest path between two vertices of $S$; (iii) variants of path covers, where one wants to select a small set of paths such that every vertex or edge belongs to one of the paths. We generalize and/or improve previous results in the area which show that the optimal values for these problems can be upper-bounded by a linear function of the cyclomatic number and the degree~1-vertices of the graph. To this end, we develop and enhance a technique recently introduced in [C. Lu, Q. Ye, C. Zhu. Algorithmic aspect on the minimum (weighted) doubly resolving set problem of graphs, Journal of Combinatorial Optimization 44:2029--2039, 2022] and give near-optimal bounds in several cases. This solves (in some cases fully, in some cases partially) some conjectures and open questions from the literature. The method, based on breadth-first search, is of algorithmic nature and thus, all the constructions can be computed in linear time. Our results also imply an algorithmic consequence for the computation of the optimal solutions: for some of the problems, they can be computed in polynomial time for graphs of bounded cyclomatic number.