The Total Interval Number of a Graph II: Trees and Complexity

Abstract

A multiple-interval representation of a simple graph $G$ assigns each vertex a union of disjoint real intervals so that vertices are adjacent if and only if their assigned sets intersect. The total interval number $I(G)$ is the minimum of the total number of intervals used in such a representation of $G$. For triangle-free graphs, $I(G)=\C{E(G)}+t(G)$, where $t(G)$ is the minimum number of pairwise edge-disjoint trails that together contain an endpoint of each edge. This yields the NP-completeness of testing $I(G)=\C{E(G)}+1$ (even for triangle-free 3-regular planar graphs) and an alternative proof that HAMILTONIAN CYCLE is NP-complete for line graphs. It also yields a linear-time algorithm to compute $I(G)$ for trees and a characterization of the trees requiring $\C{E(G)}+t$ intervals for fixed $t$. Further corollaries include the Aigner--Andreae bound of $I(G)\le\FL{(5n-3)/4}$ for $n$-vertex trees (achieved by subdividing every edge of a star), a characterization of the extremal trees, and a shorter proof of the extremal bound $\lfloor(5m+2)/4\rfloor$ for connected graphs.