The overlap number of a graph

Abstract: 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.

• The paper introduces the overlap number of a graph, defining minimal set representations with non-containment conditions and providing closed-form characterizations.
• It establishes exact and asymptotic formulas for key graph classes, including cliques, paths, cycles, and disconnected graphs using combinatorial partitioning methods.
• Complexity analyses reveal NP-completeness for several representation and extension problems, highlighting the computational challenges in overlap graph representations.

The Overlap Number of a Graph: Summary and Implications

Introduction and Motivation

This paper develops the combinatorial concept of the overlap number of a finite, simple graph $G = (V, E)$, defined as the minimal cardinality $m$ for which every vertex $v \in V$ can be assigned a set $S_v \subseteq \{1,2,\dots, m\}$ so that adjacency corresponds to set overlap: $(u,v) \in E$ iff $S_u$ and $S_v$ intersect, but neither contains the other. Contrast with intersection representations, which relax the containment condition, shows the inherent additional complexity in overlap representations, especially for realizing non-edges.

Despite the breadth of research on intersection numbers, overlap numbers have not received comparable attention. The paper presents a foundational investigation: it establishes closed-form and asymptotic characterizations for overlap numbers on canonical graph families, investigates structural properties, and proves several related computational problems to be NP-complete.

Formal Foundations and Properties

The authors begin by providing formal definitions and basic tools:

• Overlap Representation: A set function $v \mapsto S_v$ for $v \in V$ such that $(u, v) \in E$ iff $S_u \cap S_v \neq \emptyset$ and neither $S_u \subseteq S_v$ nor $S_v \subseteq S_u$.
• Overlap Number $\varphi(G)$: The minimum size $|\cup_{v \in V} S_v|$ in such a representation.

They establish that overlap representations exist for all finite graphs, and make several key structural observations. Importantly, monotonicity under induced subgraphs is proved: $\varphi(G) \ge \varphi(H)$ for any induced subgraph $H$.

A key technical lemma clarifies the structure of containments: if two connected, non-interacting subgraphs are represented, and there exists a containment between their associated sets, then the corresponding union of sets behaves almost as a single element in further containments. This is leveraged in decomposing overlap number calculations for disconnected graphs.

Overlap Numbers for Standard Graph Classes

Cliques and Complete Multipartite Graphs

The minimum overlap representation of a clique $K_n$ fundamentally requires all sets to have mutual intersection and no containments. The minimal such families correspond to subset collections of fixed size—a classical anti-chain in extremal set theory. This is captured via Milner's theorem:

• $$\varphi(K_n) = \min \{m:\ n \leq \binom{m}{\frac{m+2}{2}}\}$$
• Asymptotically, $\varphi(K_n) \in \Theta(\log n)$

For a complete $k$-partite graph, the overlap number reduces to that of a $K_k$ on the partite sets, using a vertex multiplication argument.

Paths, Cycles, and Caterpillars

The paper establishes exact overlap numbers for several sparse graph classes:

• Paths: For $P_n$ ($n \geq 3$), $\varphi(P_n)=n$.
• Cycles: For $C_n$ ($n \geq 4$), $\varphi(C_n) = n-1$.
• Caterpillars: If the spine has $k$ vertices, $\varphi = k + 2$.

All these results follow from structural partitioning arguments and efficient constructive algorithms are provided.

Disconnected Graphs

For a graph with $k$ connected components $B_1, \ldots, B_k$, the overlap number satisfies:

• $\varphi(G) = \sum_{i=1}^k \varphi(B_i) - (k-1)$

This matches the intuition that containment relationships can be leveraged across component boundaries to merge the representation size by $(k-1)$.

Computational Hardness

The paper gives thorough complexity analyses of several representation-related decision problems:

• Overlap Extension: Given a partial overlap representation and an adjacency set to a new vertex, decide if an extension using the current universe of elements is possible. Shown NP-complete via reduction from Not-All-Equal 3SAT.
• Containment-Free Overlap Number: Deciding if a containment-free (i.e., intersection-only) overlap representation of minimal size exists is NP-complete, via reduction from Intersection Number.
• $L$-Containment Overlap Number: For any fixed $L$, deciding if a representation with at most $L$ containment pairs exists is NP-complete.
• Containment Extension: For containment (partial order) representations, the analogous extension problem is also NP-complete.

In each case, reductions are algorithmically explicit and confirm that constructing or extending minimal overlap representations or their containment-restricted analogues is computationally intractable.

Implications and Future Directions

The results have several implications:

• Provides a precise combinatorial and algorithmic characterization of the overlap number for canonical and composite graphs, connecting overlap numbers with extremal set theory.
• Establishes that, in general, the minimal overlap representation is NP-complete to compute even in rather restricted settings (extension, limited containment, etc.).
• Shows that for certain sparse or regular graphs (paths, cycles, caterpillars), optimal representations can be constructed efficiently.

Future directions include identifying nontrivial graph families with tractable overlap number computation (e.g., cographs, comparability graphs) or providing good approximation algorithms. The results further suggest that classifying graphs by "overlap-complexity" may be useful, given the strong dichotomy between tractable and intractable cases even for relatively simple graph classes.

Conclusion

This work rigorously develops the theory of overlap representations of graphs, giving explicit formulas and bounds for overlap numbers on central graph classes, identifying sharp algorithmic barriers to various representation and extension problems, and connecting the area deeply with classical results in extremal set theory and intersection graph algorithms. The complexity-theoretic results, in particular, delimit the boundaries for feasible computation, and the explicit constructions for paths and related structures offer a concrete toolkit for applications where overlap representations are natural. The open problem of the complexity of the minimal overlap number computation for general graphs remains, but the results herein provide a thorough foundation and map clear routes for further investigation.