Optimal Embedding of a Tree into an Interval Graph in Linear Time

Abstract

This chapter describes the optimal embedding of a tree into an interval graph in linear time. The graphs considered in the chapter are finite, undirected, and simple. An interval supergraph for a given graph is called “optimal” if its clique number is smallest possible. The interval thickness θ ( G ) of a graph G is the minimum over the clique numbers of all interval graphs having a subgraph G. In the chapter, the following theorems are proved: (1) for any fixed integers k and d there is an algorithm deciding in time O ( n k ) for a given graph with maximal degree at most d , whether its interval thickness is at most k , and (2) For any integer k ≥ 2 and any tree T holds: θ ( T ) ≥ k + 1exists a node t V ( T ) with at least three branches T tv , T tu , T tw at t such that θ ( T tv ) ≥ k , θ ( T tu ) ≥ k , θ ( T tw ) ≥ k .