On interval representations of graphs

Abstract

The interval number i(G) of a graph G is the least integer i such that G is the intersection graph of sets of at most i intervals of the real line. The local track number l(G) is the least integer l such that G is the intersection graph of sets of at most l intervals of the real line and such that two intervals of the same vertex belong to different components of the interval representation. The track number t(G) is the least integer t such that E(G) is the union of t interval graphs. We show that the local track number of a planar graph with girth at least 7 is at most 2. We also answer a question of West and Shmoys in 1984 by showing that the recognition of 2-degenerate planar graphs with maximum degree 5 and interval number at most 2 is NP-complete.