Enumerating minimal solution sets for metric graph problems

TL;DR

This work initiates the enumeration of minimal solution sets for key metric-graph problems by connecting them to Trans-Enum, the fundamental problem of listing minimal transversals in hypergraphs. It establishes that minimal resolving sets in general graphs are equivalent to Trans-Enum, while minimal geodetic sets on split graphs are also equivalent to Trans-Enum, and shows that minimal strong resolving sets admit polynomial-delay enumeration. The paper also studies the impact of graph structure, proving hardness results that Trans-Enum underpins the difficulty of MinResolving and MinGeodetic in general, yet yielding linear-delay algorithms for P4-free graphs via Courcelle’s theorem and bounded clique-width. Additionally, it examines extension problems (Ext-Trans-Enum) and demonstrates hardness results for Ext-MinGeodetic on co-bipartite graphs and Ext-MinResolving on split graphs, highlighting limitations of standard flashlight-search methods. Overall, the work positions Trans-Enum as a central benchmark for enumeration in graph theory, while offering new insights through matroid perspectives and targeted restrictions that delineate tractable vs. intractable instances.

Abstract

Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to enumerating minimal transversals in hypergraphs (denoted Trans-Enum), whose solvability in total-polynomial time is one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in recent works: for which vertex (or edge) set graph property $Π$ is the enumeration of minimal (or maximal) subsets satisfying $Π$ equivalent to Trans-Enum? As very few properties are known to fit within this context -- namely, those related to minimal domination -- our results make significant progress in characterizing such properties, and provide new angles to approach Trans-Enum. In contrast, we observe that minimal strong resolving sets can be enumerated with polynomial delay. Additionally, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show both positive and negative results related to the enumeration and extension of partial solutions.

Figures (4)

• Figure 1: Illustration of the reduction from Trans-Enum to MinResolving with $\mathcal{H}$ consisting of $E_1=\{v_1,v_2\}$, $E_2=\{v_2,v_3,v_4\}$, $E_3=\{v_3,v_5\}$, and $E_4=\{v_4,v_5,v_6,v_7,v_8\}$. Dashed lines represent non-edges, and a bold line between two sets of vertices $A,B$ means that $A$ is complete to $B$. For legibility, we do not show the edges of $G[U]$, which is almost a clique, nor the edges of the cliques $H$, $H'$, and $V$. We only show the non-edges between $V$ and $H$. We also do not fully represent some of the edges incident to the vertices $u'_i$ and $w_j'$. The set of white vertices is one of the $O(nm)$ minimal resolving sets associated to the minimal transversal $\{v_1,v_3,v_5\}$ of $\mathcal{H}$. The set of square vertices is one of the $O(nm^2)$ minimal resolving sets not associated with a minimal transversal.
• Figure 2: Illustration of the reduction from Trans-Enum to MinGeodetic with $\mathcal{H}$ consisting of $E_1=\{v_1,v_2\}$, $E_2=\{v_2,v_3,v_4\}$, $E_3=\{v_3,v_5\}$, and $E_4=\{v_4,v_5,v_6\}$. Dashed lines represent non-edges and the bold lines incident to $u^*$ and $e^*$ mean these two vertices are complete to $I$ and $V$, respectively. For legibility, we do not represent the edges of the clique $K$. The square vertices belong to any geodetic set. The set of white vertices is a minimal geodetic set obtained from the minimal transversal $\{v_1,v_3,v_5\}$ of $\mathcal{H}$.
• Figure 3: Illustration of the reduction from Ext-Trans-Enum to Ext-MinGeodetic with $\mathcal{H}$ consisting of $E_1=\{v_1,v_2\}$, $E_2=\{v_2,v_3,v_4\}$, $E_3=\{v_3,v_5\}$, and $E_4=\{v_4,v_5,v_6\}$. The bold line between $c$ and $V$ mean that $c$ is complete to $V$. For legibility, we do not represent the edges of the cliques $V\cup \{b\}$ and $H\cup \{a,c\}$.
• Figure 4: Illustration of the reduction from Ext-Trans-Enum to Ext-MinResolving with $\mathcal{H}$ consisting of $E_1=\{v_1,v_2\}$, $E_2=\{v_2,v_3,v_4\}$, $E_3=\{v_3,v_5\}$, and $E_4=\{v_4,v_5,v_6,v_7,v_8\}$. The bold line represents the biclique between $H'$ and $V$. For legibility, we do not represent the edges of the cliques $H\cup H' \cup U' \cup U^*$.

Theorems & Definitions (44)

• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Lemma 3.6