[Paper Review] The Lusternik-Schnirelmann theorem for graphs

Frank Werner Josellis, Oliver Knill|arXiv (Cornell University)|Nov 5, 2012

Topological and Geometric Data Analysis28 references17 citations

TL;DR

This paper extends the classical Lusternik-Schnirelmann category theorem to finite simple graphs by introducing discrete analogues of contractibility, critical points, and cup length. It proves that the topological category tcat(G) is bounded above by the minimal number of critical points crit(G) of any injective function on G’s vertices, establishing a discrete version of the fundamental inequality in algebraic topology.

ABSTRACT

We prove the discrete Lusternik-Schnirelmann theorem telling that tcat(G) less or equal to crit(G) for a general simple graph G=(V,E). It relates the minimal number tcat(G) of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f on the vertex set V can have. We also prove that the cup length cup(G) is less or equal to tcat(G) which is valid also for any finite simple graph. If cat(G) is the minimal tcat(H) among all graphs H homotopic to G and cri(G) is the minimal crit(H) among all graphs H homotopic to G, we get a relation between three homotopy invariants: an algebraic quantity (cup), a topological quantity (cat) and an analytic quantity (cri).

Motivation & Objective

• To extend the classical Lusternik-Schnirelmann category theorem from smooth manifolds to finite simple graphs.
• To define and formalize discrete analogues of key topological concepts—contractibility, critical points, cup length, and homotopy—within graph theory.
• To establish a hierarchy of homotopy invariants: cup length ≤ category ≤ critical point count, valid for all finite simple graphs.
• To demonstrate that the discrete category tcat(G) and critical point count crit(G) are homotopy invariants, independent of graph representation.

Proposed method

• Define a graph as I-contractible in itself if there exists an injective function f on vertices such that all subgraphs S⁻(x) = {y ∈ S(x) | f(y) < f(x)} are contractible.
• Use vertex and edge deformation steps (a–d) to define I-homotopy, which is equivalent to homotopy via vertex operations alone (Chen-Yau-Yeh theorem).
• Define the topological category tcat(G) as the minimal tcat(H) over all graphs H homotopic to G, and crit(G) as the minimal crit(H) over such H.
• Introduce the cup length cup(G) as the maximal length of a nontrivial cup product in the cohomology ring of G.
• Prove the inequality chain cup(G) ≤ tcat(G) ≤ crit(G) using inductive constructions and discrete Morse-theoretic arguments.
• Use the filtration induced by an injective function f: V → ℝ to track homotopy changes at critical points, where the Euler characteristic may change.

Experimental results

Research questions

• RQ1 Can the Lusternik-Schnirelmann category theorem be generalized to finite simple graphs?
• RQ2 What are the discrete analogues of contractibility, critical points, and cup length in graph theory?
• RQ3 How do the invariants tcat(G), crit(G), and cup(G) relate in the discrete setting?
• RQ4 Is the category tcat(G) invariant under homotopy equivalence in graphs?
• RQ5 Can the minimal number of critical points of an injective function on a graph be bounded by its topological category?

Key findings

• The discrete Lusternik-Schnirelmann inequality holds: tcat(G) ≤ crit(G) for any finite simple graph G.
• The cup length cup(G) is bounded above by the topological category: cup(G) ≤ tcat(G).
• The category tcat(G) and critical point count crit(G) are homotopy invariants, meaning they are preserved under I-homotopy.
• A graph with crit(G) = 2 is a discrete sphere, with Betti numbers (1, 0, ..., 0, 1) and Euler characteristic 1 + (−1)^n.
• The smallest connected graph with category 2 is the cycle graph C₄.
• Computing tcat(G) or crit(G) is computationally hard, growing exponentially with graph size, and likely not in NP for general graphs.