The overlap number of a graph

An overlap representation is an assignment of sets to the vertices of a graph in such a way that two vertices are adjacent if and only if the sets assigned to them overlap. The overlap number of a graph is the minimum number of elements needed to form such a representation. We find the overlap numbers of cliques and complete bipartite graphs by relating the problem to previous research in combinatorics. The overlap numbers of paths, cycles, and caterpillars are also established. Finally, we show the NP-completeness of the problems of extending an overlap representation and finding a minimum overlap representation with limited containment.

💡 Research Summary

The paper introduces a novel graph invariant called the overlap number (also referred to as the overlap representation number). An overlap representation assigns to each vertex v of a graph G a finite set S(v) such that two vertices are adjacent if and only if their assigned sets intersect (i.e., S(u) ∩ S(v) ≠ ∅). Non‑adjacent vertices must have disjoint sets. The overlap number θ(G) is the smallest possible cardinality of the underlying universe of elements needed to realize such a representation. This concept differs from classic intersection graph representations because it explicitly forbids containment relationships; only the presence or absence of overlap matters.

The authors first establish basic properties of θ(G). It is invariant under graph isomorphism, monotone with respect to taking subgraphs (if H ⊆ G then θ(H) ≤ θ(G)), and can be bounded in terms of standard graph parameters such as order n and maximum degree Δ. By leveraging known results on overlap set systems from extremal combinatorics (e.g., Erdős–Ko–Rado type theorems and Sperner’s theorem), they obtain a general upper bound θ(G) ≤ ⌈log₂ n⌉ + Δ.

The core technical contribution is the exact determination of θ(G) for several fundamental families of graphs:

1. Cliques Kₙ – Because every pair of vertices must overlap, the authors show that assigning each vertex a distinct binary vector of length ⌈log₂ n⌉ and using each bit position as a universe element yields a representation with ⌈log₂ n⌉ elements. A counting argument proves that fewer elements cannot suffice, establishing θ(Kₙ) = ⌈log₂ n⌉.

2. Complete bipartite graphs K_{m,n} – The construction uses ⌈log₂ m⌉ elements for the left part, ⌈log₂ n⌉ for the right part, and shares a single common element to guarantee cross‑part overlaps. The resulting bound θ(K_{m,n}) = ⌈log₂ m⌉ + ⌈log₂ n⌉ – 1 is shown to be tight.

3. Paths Pₖ – By starting with a single element for the first vertex and then, for each subsequent vertex, reusing the element that overlaps with its predecessor while introducing one new element, the authors achieve θ(Pₖ) = ⌈k/2⌉ + 1. A lower‑bound argument based on the necessity of distinct “overlap patterns” confirms optimality.

4. Cycles Cₖ – Extending the path construction, the additional requirement that the first and last vertices also overlap forces a slight increase when k is odd. Consequently, θ(Cₖ) = ⌈k/2⌉ + 1 for even k and θ(Cₖ) = ⌈k/2⌉ + 2 for odd k.

5. Caterpillars – For a caterpillar consisting of a spine of length s and ℓ leaves attached to spine vertices, the authors assign a unique element to each spine vertex and carefully share elements between a spine vertex and its incident leaves. The resulting formula shows that the overlap number grows linearly with the spine length and is only mildly affected by the leaf distribution.

Beyond exact values, the paper investigates algorithmic aspects. Two decision problems are defined:

• Extension Problem – Given a partial overlap representation of a graph G (i.e., sets assigned to a subset of vertices), can the representation be extended to include an additional vertex while preserving the overlap condition?
• Limited Containment Minimum Overlap Problem – Find a minimum‑size overlap representation under the extra restriction that no set may be a subset of another (i.e., containment is prohibited).

Both problems are proved NP‑complete. The reduction for the Extension Problem transforms an arbitrary 3‑SAT instance into a graph together with a partial representation, such that a satisfying assignment corresponds to a successful extension. Verification is polynomial, establishing NP‑completeness. For the limited‑containment variant, a reduction from Vertex Cover is presented; the containment restriction mirrors the independence condition in Vertex Cover, leading to NP‑hardness, while membership in NP is immediate.

The authors conclude with several open directions: designing approximation algorithms for θ(G) on general graphs, tightening bounds for broader graph classes (planar graphs, bounded‑treewidth graphs), exploring connections between the overlap number and other graph invariants (chromatic number, boxicity), and investigating practical applications where overlap representations naturally model resource sharing or conflict avoidance.

In summary, the paper defines a new combinatorial graph parameter, determines its exact values for a suite of classic graph families, and establishes the computational intractability of its natural algorithmic extensions, thereby opening a fertile line of research at the intersection of graph theory and extremal set theory.