The total interval number of a graph, III: Tree-like graphs

Abstract

A multi-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 $I\left(G\right)$ is the minimum of the total number of intervals used in such a representation of $G$. We present a linear-time algorithm to compute $I\left(G\right)$ when every block of $G$ is a complete graph or a cycle. We apply the algorithm to prove extremal results. For an $n$-vertex cactus (every block is an edge or a cycle), the maximum of $I\left(G\right)$ is $\left((18n-12)/13\right)$. For an $n$-vertex block graph (every block is a complete graph), the maximum is $\left(3n/2-2\right)$. In both extremal results there are a few small exceptions.