General Position (GP) Task in Graph Theory

• General Position (GP) Task is defined as finding a maximal vertex subset where no three vertices lie on a common shortest path, unifying geometric and combinatorial concepts.
• Techniques such as isometric covers, BFS-tree leaf counts, and packings provide upper and lower bounds, offering practical strategies for various graph families.
• The GP Task is NP-complete in general graphs, with tractable cases in trees and block graphs, making it a pivotal open problem in metric graph combinatorics.

The general position (GP) problem in graph theory extends the classical Euclidean concept of general position—where no three points are collinear—to the context of graph geodesics: a set of vertices is in general position if no three lie on a common shortest path. The primary quantity of interest, the general position number $\gp(G)$, is the size of a maximum cardinality subset in general position. This problem unifies metric subgraph properties, extremal combinatorics, simpliciality concepts, and connects to the complexity of classic packing and covering tasks.

1. Formal Definitions

Let $G = (V, E)$ be a connected graph.

• General position set: $S \subseteq V(G)$ is in general position if no three vertices of $S$ lie on a common geodesic, i.e., there do not exist distinct $x, y, z \in S$ such that the shortest path from $x$ to $z$ passes through $y$.
• General position number: The general position number $\gp(G)$ is the largest cardinality of a general position set in $G$,

$\gp(G) = \max\{\,|S| : S \subseteq V(G) \text{ is in general position}\,\}.$

Geodesics here refer exclusively to shortest paths. The problem is thus the search for maximal sets avoiding “collinear” configurations in the metric space induced by $G$ (Manuel et al., 2017).

2. Upper Bounds: Isometric Covers and Related Techniques

A core paradigm for upper bounding $\gp(G)$ is via isometric covers—collections of isometric subgraphs (distance-preserving subgraphs for all pairs of their vertices) whose vertex-sets collectively cover $V(G)$.

• Isometric Cover Lemma: For any isometric cover $\{H_1, \cdots, H_k\}$ of $G$,

$\gp(G) \leq \sum_{i=1}^k \gp(H_i).$

This follows since any general position set $S$ intersects each $H_i$ in a general position set of $H_i$.

• Immediate corollaries:
  • $\gp(G) \leq 2\,\ip(G)$, where $\ip(G)$ is the minimum number of isometric paths covering $G$.
  • $\gp(G) \leq 3\,\ic(G)$, where $\ic(G)$ is the minimum isometric cycle cover.
  • For a vertex $v$, $\gp(G) \leq \min_{v \in V}\;1 + \ip(v, G)$, where $\ip(v, G)$ is the minimum number of isometric paths rooted at $v$ needed to cover $V(G)$. For BFS-trees with $\ell(v)$ leaves, $\ip(v, G) \leq \ell(v)$, yielding $\gp(G) \leq 1 + \min\{\ell(v)\}$.

These bounds allow for reductions on product graphs, many chordal families, and for obtaining explicit general position number estimates on finite grids or trees (Manuel et al., 2017).

3. Lower Bounds: Packings and Simplicial Vertices

Lower bounds are provided by exploiting specific graph structures.

• Simplicial vertices: A vertex is simplicial if its closed neighborhood induces a clique. The set of all simplicial vertices forms a general position set—no three simplicial vertices can be collinear, due to maximal connectivity.

$\gp(G) \geq |\text{simplicial vertices of } G|$

• Packing-Diameter Bound: For $k \geq 1$, a $k$-packing is a set of vertices pairwise at distance $>k$, with size $\alpha_k(G)$.
  • When $\operatorname{diam}(G) \leq 2k+1$, any $k$-packing must be in general position, so $\gp(G) \geq \alpha_k(G)$.
• Edge-distance Bound: If $F$ is a set of edges pairwise at distance exactly $\operatorname{diam}(G)$, their endpoints form a general position set, so $\gp(G)\geq 2|F|$.

These criteria are particularly effective in chordal graphs and block families (Manuel et al., 2017).

4. Exact Values and Tight Bounds on Graph Families

The GP-number is known exactly or up to tight bounds for several canonical graph classes:

Graph Family	General Position Number ($\gp(G)$)	Additional Notes
Path $P_n$	$2$ (for all $n \geq 2$)
Cycle $C_n$	$2$ for $n=4$; $3$ for $n \neq 4$
Complete graph $K_n$	$n$
Tree $T$	$|L|$ (number of leaves)
Block graphs	#simplicial vertices
Theta graphs $\Theta(k,\ell)$	$k+1$
Glued binary trees	$2^r$ (depth $r$)
Petersen graph	$6$	via isometric-cycle cover/packing

For grid graphs, only upper and lower bounds are available; no closed-form solution exists (Manuel et al., 2017).

5. Complexity and Algorithms

The decision problem “Is $\gp(G)\geq k$?” is NP-complete, via reduction from Maximum Independent Set. The construction involves replacing each vertex with a “gadget”—a clique and a pendant—which preserves the relationship between independent sets and general position sets in the augmented graph.

For certain classes such as trees and block graphs, polynomial-time exact algorithms are available. For general graphs, no polynomial-time or constant-factor approximation algorithms are known. Greedy heuristics—favoring vertices with minimal isometric path cover or maximizing distance to the current set—are used in practice, though no rigorously proven approximation guarantees exist (Manuel et al., 2017).

6. Structural Insights, Open Problems, and Practical Strategies

The GP-task is fundamentally tied to classical metric invariants ($\ip(G)$, $\ic(G)$, packing numbers), structural subgraph decomposition (simplicial vertices, BFS-leaf distribution), and block structure. In practice:

• For upper bounds: Identify minimal isometric covers, use BFS-tree leaf counting, and exploit block or chordal decompositions.
• For lower bounds: Locate all simplicial vertices, find large packings, and use edge-distance criteria.
• The problem is solved for trees, block graphs, cographs (via modular decomposition), theta graphs, and select cacti/hose graphs.

Open combinatorial problems include determining the exact GP-number for grids, hypercubes, toroidal grids, and general bipartite planar graphs, as well as developing efficient approximation or parameterized algorithms for computing $\gp(G)$ (Manuel et al., 2017).

7. Applications and Broader Context

The general position problem in graphs abstracts geometric no-three-in-line motifs to metric discrete settings. It acts as a benchmark for metric packing and covering techniques, links to geometric configurations in combinatorics, and may be relevant in domains such as sensor network placement (to avoid degenerate collinear alignments), error-correcting code constructions, and algorithmic geometry. The metric and combinatorial interplay suggests further connections to graph isometric embedding and extremal network design.

The general position problem remains a central unsolved challenge in metric graph combinatorics, with substantial ties to both structural graph theory and classical geometry (Manuel et al., 2017).