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

2025-08-30

AI Summary

This paper investigates distance- and path-based covering problems—including metric dimension, geodetic sets, and path cover—on graphs with bounded cyclomatic number. We develop a unified structural approach that synergistically handles degree-one vertices and cycle structures via breadth-first search and graph-theoretic analysis, yielding linear upper bounds on solution sizes dependent solely on the cyclomatic number and leaf count. This work constitutes the first systematic extension of such covering problems to the class of bounded-cyclomatic-number graphs, improving several existing upper bounds and partially resolving long-standing open conjectures from the literature. Our algorithms are constructive in linear time and solvable in polynomial time, achieving near-optimal bounds across multiple problem variants. The framework establishes a novel paradigm for distance-related optimization on sparse graphs, bridging structural graph theory with combinatorial optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Studying graph covering problems based on distances and paths
• Focusing on graphs with bounded cyclomatic number constraints
• Generalizing metric dimension, geodetic sets, and path covers

Innovation

• Uses breadth-first search for algorithmic graph analysis
• Bounds solutions linearly with cyclomatic number
• Enhances resolving set technique for near-optimal results