Overlap Number of Graphs

TL;DR

This paper introduces the concept of overlap number for graphs, providing bounds and algorithms for specific classes such as trees, planar graphs, and general graphs, with proofs of optimality and computational efficiency.

Contribution

The paper establishes new bounds on the overlap number for various graph classes and provides linear-time algorithms for optimal representations of trees.

Findings

Optimal overlap representation of trees can be found in linear time.

Bounds on overlap number for graphs with minimum degree at least 2.

Upper bounds on overlap number for planar graphs and general graphs, with sharpness proofs.

Abstract

An {\it overlap representation} of a graph $G$ assigns sets to vertices so that vertices are adjacent if and only if their assigned sets intersect with neither containing the other. The {\it overlap number} $\ol(G)$ (introduced by Rosgen) is the minimum size of the union of the sets in such a representation. We prove the following: (1) An optimal overlap representation of a tree can be produced in linear time, and its size is the number of vertices in the largest subtree in which the neighbor of any leaf has degree 2. (2) If $\delta(G)\ge 2$ and $G\ne K_3$, then $\ol(G)\le |E(G)|-1$, with equality when $G$ is connected and triangle-free and has no star-cutset. (3) If $G$ is an $n$-vertex plane graph with $n\ge5$, then $\ol(G)\le 2n-5$, with equality when every face has length 4 and there is no star-cutset. (4) If $G$ is an $n$-vertex graph with $n\ge 14$, then $\ol(G)\le \floor{n^2/4-n/2-1}$, and this is sharp (for even $n$, equality holds when $G$ arises from $K_{n/2,n/2}$ by deleting a perfect matching).